12. Multiple and Submultiple Angles

12.1. Multiple Angles

An angle of the form \(nA,\) where \(n\) is an integer is called a multiple angle. For example, \(2A, 3A, 4A, \ldots\) are multiple angles of \(A.\)

12.1.1. Trigonometrical Ratios of \(2A\)

From previous chapter we know that \(\sin(A + B) = \sin A\cos B + \cos A\sin B\)

Substituting \(B = A,\) we get

\(\sin 2A = 2\sin A\cos A\)

Similarly, \(\cos 2A = \cos^2A - \sin^2A = 2\cos^2A - 1 = 1 - 2\sin^2A\) (recall formula from previous chapter and substitute \(B = A\) \(\cos^2A = 1 -\sin^2A\) and \(\sin^2A = 1 - \cos^2a\))

Also, \(\tan 2A = \frac{2\tan A}{1 - \tan^2A}\) (recall formula from previous chapter and put \(B = A\))

12.1.2. \(\sin 2A\) and \(\cos 2A\) in terms of \(\tan A\)

\(\sin 2A = \frac{2\sin A\cos A}{\sin^2A + \cos^2A}[\because \sin^2A + \cos^2A = 1]\)

Dividing both numerator and denominator by \(\cos^2A,\) we get

\(\sin 2A = \frac{2\tan A}{1 + \tan^2A}\)

\(\cos A = \cos^2A - \sin^2A = \frac{\cos^2A - \sin^2A}{\cos^2A + \sin^2A}[\because \sin^2A + \cos^2A = 1]\)

Dividing both numerator and denominator by \(\cos^2A,\) we get

\(\cos 2A = \frac{1 - \tan^2A}{1 + \tan^2A} = \frac{\cot^2A - 1}{\cot^2A + 1}\)

12.1.3. Trigonometrical Ratios of \(3A\)

\(\sin 3A = \sin2A\cos A + \cos 2A\sin A = 2\sin A\cos^2 A + \cos^2A\sin A - \sin^3A\)

\(= 2\sin A(1 - \sin^2A) + (1 - 2\sin^2A)\sin A - \sin^3A\)

\(= 3\sin A - 4\sin^3A\)

\(\cos 3A = \cos2A\cos A - \sin 2A\sin A = (2\cos^2A - 1)\cos A - 2\sin^2 A\cos A\)

\(= 2\cos^3A - \cos A - 2(1 - \cos^2A)\cos A\)

\(= 4\cos^3 A - 3\cos A\)

We know that \(\tan(A + B + C) = \frac{\tan A + \tan B + \tan C - \tan A\tan B\tan C}{1 - \tan A\tan B - \tan B\tan C - \tan C\tan A}\)

Putting \(B = A\) and \(C = A,\) we get

\(\tan 3A = \frac{3\tan A - \tan^3A}{1 - 3\tan^2A}\)

Similarly \(\cot 3A = \frac{\cot^3 A - 3\cot A}{3\cot^2A - 1}\)

12.1.4. Some Important Formulae

  1. \(\cos2A = 1 - 2\sin^2A \Rightarrow \sin^2A = \frac{1}{2}(1 - \cos2A)\)

  2. \(\cos2A = 2\cos^2 - 1 \Rightarrow \cos^2A = \frac{1}{2}(1 + \cos2A)\)

  3. \(\sin 3A = 3\sin A - 4\sin^3A \Rightarrow sin^3A = \frac{1}{2}(3\sin A - \sin3A)\)

  4. \(\cos 3A = 4\cos^3A - 3\cos A \Rightarrow \cos^3A = \frac{1}{4}(\cos3A + 3\cos A)\)

12.2. Submultiple Angles

An angle of the form \(\frac{A}{n},\) where \(n\) is an integer is called a submultiple angle. For exmaple, \(\frac{A}{2}, \frac{A}{3}, \frac{A}{4}, \ldots\) are submultiple angles of \(A.\)

12.2.1. Trigonometrical Ratios of \(A/2\)

We know that, \(\sin 2A = 2\sin A\cos A.\) Putting \(A=A/2,\) we get

\(\sin A = 2\sin A/2\cos A/2\)

\(\cos 2A = \cos^2A - \sin^A.\) Putting \(A = A/2,\) we get

\(\cos A - \cos^2\frac{A}{2} - \sin^2\frac{A}{2}\)

\(\cos 2A = 2\cos^2A - 1.\) Putting \(A = A/2,\) we get

\(\cos A = 2\cos^2\frac{A}{2} - 1\)

\(\cos 2A = 1 - 2\sin^2A.\) Putting \(A = A/2,\) we get

\(\cos A = 1 - 2\sin^2\frac{A}{2}\)

\(\tan 2A = \frac{2\tan A}{1 - \tan^2A}.\) Putting \(A = A/2,\) we get

\(\tan A = \frac{2\tan \frac{A}{2}}{1 - \tan^2\frac{A}{2}}\)

\(\sin 2A = \frac{2\tan A}{1 + \tan^2A} \therefore \sin A = \frac{2\tan \frac{A}{2}}{1 + \tan^2\frac{A}{2}}\)

\(\cos 2A = \frac{1 - \tan^2A}{1 + \tan^2A} \therefore \cos A = \frac{1 - \tan^2\frac{A}{2}}{1 + \tan^2\frac{A}{2}}\)

\(\cot 2A = \frac{\cot^2A - 1}{2\cot A} \therefore \cot A = \frac{\cot^2\frac{A}{2} - 1}{2\cot \frac{A}{2}}\)

12.2.2. Trigonometrical Ratios of \(A/3\)

\(\sin 3A = 3\sin A - 4\sin^3A.\) Putting \(A = \frac{A}{3},\) we get

\(\sin A = \frac{3}{\sin \frac{A}{3}} - 4\sin^3\frac{A}{3}\)

\(\cos 3A = 4\cos^3A - 3\cos A\). Putting \(A = \frac{A}{3},\) we get

\(\cos A = 4\cos^3\frac{A}{3} - 3\cos \frac{A}{3}\)

\(\tan 3A = \frac{3\tan A - \tan^3A}{1 - 3\tan^2A}\)

\(\tan A = \frac{3\tan\frac{A}{3} - \tan^3\frac{A}{3}}{1 - 3\tan^2\frac{A}{3}}\)

12.2.3. Values of \(cos A/2, \sin A/2\) and \(\tan A/2\) in terms of \(\cos A\)

\(\cos^2\frac{A}{2} = \frac{1 + \cos A}{2} \therefore \cos \frac{A}{2} = \sqrt{\frac{1 + \cos A}{2}}\)

\(\sin^2\frac{A}{2} = \frac{1 - \cos A}{}2 \therefore \sin \frac{A}{2} = \sqrt{\frac{1 - \cos A}{2}}\)

\(\tan^2\frac{A}{2} = \frac{1 - \cos A}{1 + \cos A} \therefore \tan\frac{A}{2} = \sqrt{\frac{1 - \cos A}{1 + \cos A}}\)

12.2.4. Values of \(\sin A/2\) and \(\cos A/2\) in terms of \(\sin A\)

\(\left(\cos \frac{A}{2} + \sin\frac{A}{2}\right)^2 = \cos^2\frac{A}{2} + \sin^2\frac{A}{2} + 2\cos\frac{A}{2}\sin\frac{A}{2}\)

\(= 1 + \sin A \Rightarrow \cos \frac{A}{2} + \sin \frac{A}{2} = \sqrt{1 + \sin A}\)

Similarly, \(\cos \frac{A}{2} - \sin \frac{A}{2} = \sqrt{1 - \sin A}\)

Adding, we get \(\cos \frac{A}{2} = \pm\frac{1}{2}\sqrt{1 + \sin A} \pm\frac{1}{2}\sqrt{1 - \sin A}\)

Subtracting, we get \(\cos \frac{A}{2} = \pm\frac{1}{2}\sqrt{1 + \sin A} \mp\frac{1}{2}\sqrt{1 - \sin A}\)

12.2.5. Value of \(\sin 18^\circ\) and \(\cos 72^\circ\)

Let \(A = 18^\circ,\) then \(\sin 5A = 90^\circ \therefore 2A + 3A = 90^\circ\)

\(\sin2A = \sin(90^\circ - \sin 3A) \therefore 2\sin A\cos A = 4\cos^3A - 3\cos A\)

Dividing both sides by \(\cos A,\) we get

\(2\sin A = 4\cos^2A - 3 = 4(1 - \sin^2A) - 3\)

\(4\sin^2A + 2\sin A - 1 = 0\)

\(\sin A = \frac{-1\pm\sqrt{5}}{4}\)

However, since \(A= 18^\circ\therefore \sin A > 0\)

\(\therefore \sin18^\circ = \frac{-1 + \sqrt{5}}{4}\)

\(\therefore \sin(90^\circ - 18^\circ) = \cos72^\circ = \frac{\sqrt{5} - 1}{4}\)

12.2.6. Value of \(\cos 18^\circ\) and \(\sin 72^\circ\)

\(\cos^218^\circ = 1 - \sin^218^\circ = 1 - \left(\frac{\sqrt{5} - 1}{4}\right)^2\)

\(= \frac{10 + 2\sqrt{5}}{16}\therefore \cos18^\circ = \frac{1}{4}\sqrt{10 + 2\sqrt{5}}[\because \cos18^\circ > 0]\)

\(\cos(90^\circ - 18^\circ) = \sin72^\circ = \frac{1}{4}\sqrt{10 + 2\sqrt{5}}\)

12.2.7. Value of \(\tan 18^\circ\) and \(\tan 72^\circ\)

\(\tan 18^\circ = \frac{\sin18^\circ}{\cos18^\circ} = \frac{\sqrt{5} - 1}{\sqrt{10 + 2\sqrt{5}}}\)

\(\tan18^\circ\cot18^\circ = 1\Rightarrow \tan72^\circ = \frac{1}{\tan18^\circ} = \frac{\sqrt{10 + 2\sqrt{5}}}{\sqrt{5} - 1}\)

12.2.8. Value of \(\cos 36^\circ\) and \(\sin 54^\circ\)

\(\cos 36^\circ = 1 - 2\sin^218^\circ = 1 - 2\left(\frac{\sqrt{5} - 1}{4}\right)^2\)

\(= \frac{\sqrt{5} + 1}{4}\)

\(\sin 54^\circ = \sin(90^\circ - 36^\circ) = \cos36^\circ = \frac{\sqrt{5} + 1}{4}\)

12.2.9. Value of \(\sin 36^\circ\) and \(\cos 54^\circ\)

\(\sin36^\circ = 1 - \cos^236^\circ = 1 - \left(\frac{\sqrt{5} + 1}{4}\right)^2\)

\(= \frac{1}{4}\sqrt{10 - 2\sqrt{5}}\)

\(\cos54^\circ = \cos(90^\circ - 36^\circ) = \sin36^\circ = \frac{1}{4}\sqrt{10 - 2\sqrt{5}}\)

Several other angles like, \(9^\circ, 15^\circ, 22\frac{1}{2}^\circ, 7\frac{1}{2}^\circ\) etc can be found similarrly.

12.3. Problems

  1. Find the value of \(\sin 2A,\) when

    1. \(\cos A = \frac{3}{5}\)

    2. \(\sin A = \frac{12}{13}\)

    3. \(\tan A = \frac{16}{63}\)

  2. Find the value of \(\cos 2A,\) when

    1. \(\cos A = \frac{15}{17}\)

    2. \(\sin A = \frac{4}{5}\)

    3. \(\tan A = \frac{5}{12}\)

  3. If \(\tan A = \frac{b}{a},\) find the value of \(a\cos 2A+ b\sin 2A\)

Prove that

  1. \(\frac{\sin 2A}{1 + \cos 2A} = \tan A\)

  2. \(\frac{\sin 2A}{1 - \cos 2A} = \cot A\)

  3. \(\frac{1 - \cos 2A}{1 + \cos 2A} = \tan^2A\)

  4. \(\tan A + \cot A = 2\cosec 2A\)

  5. \(\tan A - \cot A = -2\cot2A\)

  6. \(\cosec 2A + \cot 2A = \cot A\)

  7. \(\frac{1 - \cos A + \cos B - \cos(A + B)}{1 + \cos A - \cos B - \cos(A + B)} = \tan\frac{A}{2}\cot\frac{B}{2}\)

  8. \(\frac{\cos A}{1 \mp \sin A} = \tan\left(45^\circ \pm \frac{A}{2}\right)\)

  9. \(\frac{\sec 8A - 1}{\sec 4A - 1} = \frac{\tan 8A}{\tan 2A}\)

  10. \(\frac{1 + \tan^2(45^\circ - A)}{1 - \tan^2(45^\circ - A)} = \cosec 2A\)

  11. \(\frac{\sin A + \sin B}{\sin A - \sin B} = \frac{\tan \frac{A + B}{2}}{\tan \frac{A - B}{2}}\)

  12. \(\frac{\sin^2A - \sin^2B}{\sin A\cos A - \sin B\cos B} = \tan(A + B)\)

  13. \(\tan\left(\frac{\pi}{4} + A\right) - \tan\left(\frac{\pi}{4} - A\right) = 2\tan 2A\)

  14. \(\frac{\cos A + \sin A}{\cos A - \sin A} - \frac{\cos A - \sin A}{\cos A + \sin A} = 2\tan 2A\)

  15. \(\cot (A + 15^\circ) - \tan(A - 15^\circ) = \frac{4\cos 2A}{1 + 2\sin 2A}\)

  16. \(\frac{\sin A + \sin2A}{1 + \cos A + \cos 2A} = \tan A\)

  17. \(\frac{1 + \sin A - \cos A }{1 + \sin A + cos A} = \tan \frac{A}{2}\)

  18. \(\frac{\sin(n + 1)A - \sin(n - 1)A}{\cos(n + 1)A + 2\cos nA + \cos(n - 1)A} = \tan \frac{A}{2}\)

  19. \(\frac{\sin(n + 1)A + 2\sin nA + \sin(n - 1)A}{\cos(n - 1) - \cos(n + 1)A} = \cot \frac{A}{2}\)

  20. \(\sin(2n + 1)A\sin A = \sin^2(n + 1)A - \sin^2nA\)

  21. \(\frac{\sin(A + 3B) + \sin(3A + B)}{\sin 2A + \sin 2B} = 2\cos(A + B)\)

  22. \(\sin 3A + \sin 2A - \sin A = 4\sin A\cos \frac{A}{2}\cos \frac{3A}{2}\)

  23. \(\tan 2A = (\sec 2A + 1)\sqrt{\sec^2A - 1}\)

  24. \(\cos^32A + 3\cos 2A = 4(\cos^6A - \sin^6A)\)

  25. \(1 + \cos^22A = 2(\cos^4A + \sin^4A)\)

  26. \(\sec^2A(1 + \sec2A) = 2\sec2A\)

  27. \(\cosec A - 2\cot 2A\cos A = 2\sin A\)

  28. \(\cot A = \frac{1}{2}\left(\cot\frac{A}{2} - \tan\frac{A}{2}\right)\)

  29. \(\sin A\sin(60^\circ - A)\sin(60^\circ + A) = \frac{1}{4}\sin 3A\)

  30. \(\cos A\cos(60^\circ - A)\cos(60^\circ + A) = \frac{1}{4}\cos 3A\)

  31. \(\cot A + \cot(60^\circ + A) - \cot(60^\circ - A) = 3\cot 3A\)

  32. \(\cos 4A = 1 - 8\cos^2A + 8\cos^4A\)

  33. \(\sin 4A = 4\sin A\cos^3A - 4\cos A\sin^3A\)

  34. \(\cos 6A = 32\cos^6A - 48\cos^4A + 18\cos^2A - 1\)

  35. \(\tan 3A\tan 2A\tan A = \tan 3A - \tan 2A - \tan A\)

  36. \(\frac{2\cos2^nA + 1}{2\cos A + 1} = (2\cos A - 1)(2\cos 2A - 1)(2\cos2^2A - 1)\ldots(2\cos2^{n - 1} - 1)\)

  37. If \(\tan A= \frac{1}{7}, \sin B = \frac{1}{\sqrt{10}},\) prove that \(A + 2B = \frac{\pi}{4},\) where \(0 < A < \frac{\pi}{4}\) and \(0 < B < \frac{\pi}{4}\)

Prove that

  1. \(\tan\left(\frac{\pi}{4} + A\right) + \tan\left(\frac{\pi}{4} - A\right) = 2\sec2A\)

  2. \(\sqrt{3}\cosec 20^\circ - \sec 20^\circ = 4\)

  3. \(\tan A + 2\tan 2A + 4\tan 4A + 8\cot 8A = \cot A\)

  4. \(\cos^2A + \cos^2\left(\frac{2\pi}{3} - A\right) + \cos^2\left(\frac{2\pi}{3} + A\right) = \frac{3}{2}\)

  5. \(2\sin^2A + 4\cos (A + B)\sin A\sin B + \cos2(A + B)\) is idnependent of \(A.\)

  6. If \(\cos A = \frac{1}{2}\left(a + \frac{1}{a}\right),\) show that \(\cos 2A = \frac{1}{2}\left(a^2 + \frac{1}{a^2}\right)\)

    Prove that

  7. \(\cos^2A + \sin^2A\cos 2B = \cos^2B + \sin^2B\cos 2A\)

  8. \(1 + \tan A\tan 2A = \sec 2A\)

  9. \(\frac{1 + \sin 2A}{1 - \sin 2A} = \left(\frac{1 + \tan A}{1 - \tan A}\right)^2\)

  10. \(\frac{1}{\sin 10^\circ} - \frac{\sqrt{3}}{\cos 10^\circ} = 4\)

  11. \(\cot^2A - \tan^2A = 4\cot2A\cosec 2A\)

  12. \(\frac{1 +\sin 2A}{\cos2A} = \frac{\cos A + \sin A}{\cos A - \sin A} = \tan\left(\frac{\pi}{4} + A\right)\)

  13. \(\cos^6A - \sin^6A = \cos2A\left(1 - \frac{1}{4}\sin^22A\right)\)

  14. \(\cos^2A + \cos^2\left(\frac{\pi}{3} + A\right) + \cos^2\left(\frac{\pi}{3} - A\right)= \frac{3}{2}\)

  15. \((1 + \sec2A)(1 + \sec2^2A)(1 + \sec2^3A) \ldots (1 + \sec2^nA) = \frac{\tan2^nA}{\tan A}\)

  16. \(\frac{\sin2^nA}{\sin A} = 2^n\cos A\cos 2A\cos 2^2A\ldots\cos2^{n - 1}A\)

  17. \(3(\sin A - \cos A)^4 + 6(\sin A + \cos A)^2 + 4(\sin^6A + \cos^6A) = 13\)

  18. \(2(\sin^6A + \cos^6A) - 3(\sin^4A + \cos^4A) + 1 = 0\)

  19. \(\cos^2A + \cos^2(A + B) -2\cos A\cos B\cos(A + B)\) if independent of \(A.\)

  20. \(\cos^3A\cos 3A + \sin^3A\sin 3A = \cos^32A\)

  21. \(\tan A\tan(60^\circ - A)\tan(60^\circ + A) = \tan 3A\)

  22. \(\sin^2A + \sin^3\left(\frac{2\pi}{3} + A\right) + \sin^3\left(\frac{4\pi}{3} + A\right) = -\frac{3}{4}\sin 3A\)

  23. \(4(\cos^310^\circ + \sin^320^\circ) = 3(\cos 10\circ + \sin 20^\circ)\)

  24. \(\sin A\cos^3A - \cos A\sin^3A = \frac{1}{4}\sin 4A\)

  25. \(\cos^3A\sin3A + \sin^3A\cos 3A = \frac{3}{4}\sin 4A\)

  26. \(\sin A\sin(60^\circ + A)\sin(A + 120^\circ) = \sin 3A\)

  27. \(\cot A + \cot(60^\circ + A) + \cot(120^\circ + A) = 3\cot 3A\)

  28. \(\cos 5A = 16\cos^5A - 20\cos^3A + 5\cos A\)

  29. \(\sin 5A = 5\sin A - 20\sin^3A + 16\sin^5A\)

  30. \(\cos 4A - \cos 4B = 8(\cos A - \cos B)(\cos A + \cos B)(\cos A - \sin B)(\cos A + \sin B)\)

  31. \(\tan 4A = \frac{4\tan A - 4\tan^3A}{1 - 6\tan^2A + \tan^4A}\)

  32. If \(2\tan A = 3\tan B,\) prove that \(\tan (A- B) = \frac{\sin 2B}{5 - \cos 2B}\)

  33. If \(\sin A + \sin B = x\) and \(\cos A + \cos B = y,\) show that \(\sin(A + B) = \frac{2xy}{x^2 + y^2}\)

  34. If \(A= \frac{\pi}{2^n + 1},\) prove that \(\cos A.\cos 2A. \cos2^2A.\ldots.\cos2^{n - 1}A = \frac{1}{2^n}\)

  35. If \(\tan A = \frac{y}{x},\) prove that \(x\cos 2A + y\sin 2A = x\)

  36. If \(\tan^2A = 1 + 2\tan^2B,\) prove that \(\cos 2B = 1 + 2\cos 2A\)

  37. If \(A\) and \(B\) lie between \(0\) and \(\frac{\pi}{2}\) and \(\cos 2A = \frac{3\cos 2B - 1}{3 - \cos 2B},\) prove that \(\tan A = \sqrt{2}\tan B\)

  38. If \(\tan B = 3\tan A,\) prove that \(\tan(A + B) = \frac{2\sin 2B}{1 + \cos 2B}\)

  39. If \(x\sin A = y\cos A,\) prove that \(\frac{x}{\sec 2A} + \frac{y}{\cosec 2A} = x\)

  40. If \(\tan A = \sec 2B,\) prove that \(\sin 2A = \frac{1 - \tan^4B}{1 + \tan^4B}\)

  41. If \(A = \frac{\pi}{3},\) prove that \(\cos A.\cos 2A. \cos 3A.\cos 4A.\cos 5A.\cos 6A = -\frac{1}{16}\)

  42. If \(A = \frac{\pi}{15},\) prove that \(\cos2A.\cos4A.\cos8A.\cos14A = \frac{1}{16}\)

  43. If \(\tan A\tan B = \sqrt{\frac{a - b}{a + b}},\) prove that \((a - b\cos2A)(a - b\cos2B) = a^2 - b^2\)

  44. If \(\sin A = \frac{1}{2}\) and \(\sin B = \frac{1}{3},\) find the value of \(\sin(A + B)\) and \(\sin(2A + 2B)\)

  45. If \(\cos A = \frac{11}{61}\) and \(\sin B = \frac{4}{5},\) find the value of \(\sin^2 \frac{A - B}{2}\) and \(cos^2\frac{A + B}{2},\) the angle of \(A\) and \(B\) being positive acute angles.

  46. Given \(\sec A = \frac{5}{4},\) find \(\tan\frac{A}{2}\) and \(\tan A.\)

  47. If \(\cos A = .3,\) find the value of \(\tan \frac{A}{2},\) and explain the resulting ambiguity.

  48. If \(\sin A + \sin B = x\) and \(\cos A + \cos B = y,\) find the value of \(\tan \frac{A - B}{2}\)

Prove that

  1. \((\cos A + \cos B)^2 + (\sin A - \sin B)^2 = 4\cos^2 \frac{A + B}{2}\)

  2. \((\cos A + \cos B)^2 + (\sin A + \sin B)^2 = 4\cos^2 \frac{A - B}{2}\)

  3. \((\cos A - \cos B)^2 + (\sin A - \sin B)^2 = 4\sin^2 \frac{A - B}{2}\)

  4. \(\sin^2\left(\frac{\pi}{8} + \frac{A}{2}\right) - \sin^2\left(\frac{\pi}{8} -\frac{A}{2}\right) = \frac{1}{\sqrt{2}}\sin A\)

  5. \((\tan 4A + \tan 2A)(1 - \tan^23A\tan^2A) = 2\tan 3A\sec^2A\)

  6. \(\left(1 + \tan \frac{A}{2} - \sec\frac{A}{2}\right)\left(1 + \tan \frac{A}{2} + \sec\frac{A}{2}\right) = \sin A\sec^2\frac{A}{2}\)

  7. \(\frac{1 + \sin A - \cos A}{1 + \sin A + \cos A} = \tan \frac{A}{2}\)

  8. \(\frac{1 - \tan \frac{A}{2}}{1 + \tan \frac{A}{2}} = \frac{1 + \sin A}{\cos A} = \tan \left(\frac{\pi}{4} + \frac{A}{2}\right)\)

  9. \(\cos^4\frac{\pi}{8} + \cos^4 \frac{3\pi}{8} + \cos^4\frac{5\pi}{8} + \cos^4\frac{7\pi}{8}= \frac{3}{2}\)

  10. \(\frac{2\sin A - \sin2A}{2\sin A + \sin 2A} = \tan^2\frac{A}{2}\)

  11. \(\cot \frac{A}{2} - \tan \frac{A}{2} = 2\cot A\)

  12. \(\frac{1 + \sin A}{1 - \sin A} = \tan^2\left(\frac{\pi}{4} + \frac{A}{2}\right)\)

  13. \(\sec A + \tan A = \tan\left(\frac{\pi}{4} + \frac{A}{2}\right)\)

  14. \(\frac{\sin A + \sin B - \sin(A + B)}{\sin A + \sin B + \sin(A + B)} = \tan \frac{A}{2}\tan \frac{B}{2}\)

  15. \(\tan \left(\frac{\pi}{4} - \frac{A}{2}\right) = \sec A - \tan A = \sqrt{\frac{1 - \sin A}{1 + \sin A}}\)

  16. \(\cosec\left(\frac{\pi}{4} + \frac{A}{2}\right)\cosec \left(\frac{\pi}{4} - \frac{A}{2}\right) = 2\sec A\)

  17. \(\cos^2\frac{\pi}{8} + \cos^2\frac{3\pi}{8} + \cos^2\frac{5\pi}{8} + \cos^2\frac{7\pi}{8} = 2\)

  18. \(\sin^4\frac{\pi}{8} + \sin^4 \frac{3\pi}{8} + \sin^4\frac{5\pi}{8} + \sin^4\frac{7\pi}{8} = \frac{3}{2}\)

  19. \(\left(1 + \cos \frac{\pi}{8}\right)\left(1 + \cos\frac{3\pi}{8}\right)\left(1 + \cos\frac{5\pi}{8}\right)\left(1 + \cos \frac{7\pi}{8}\right) = \frac{1}{8}\)

  20. Find the value of \(\sin \frac{23\pi}{24}\)

  21. If \(A = 112^\circ30',\) find the value of \(\sin A\) and \(\cos A\)

Prove that

  1. \(\sin^224^\circ - \sin^26^\circ = \frac{1}{8}(\sqrt{5} - 1)\)

  2. \(\tan6^\circ.\tan42^\circ.\tan66^\circ.\tan78^\circ = 1\)

  3. \(\sin47^\circ + \sin61^\circ - \sin 11^\circ - \sin25^\circ = \cos 7^\circ\)

  4. \(\sin 12^\circ\sin48^\circ\sin54^\circ = \frac{1}{8}\)

  5. \(\cot 142\frac{1}{2}^\circ = \sqrt{2} + \sqrt{3} - 2 - \sqrt{6}\)

  6. \(\sin^248^\circ - \cos^212^\circ = -\frac{\sqrt{5} + 1}{8}\)

  7. \(4(\sin 24^\circ + \cos6^\circ) = \sqrt{3} + \sqrt{15}\)

  8. \(\cot6^\circ\cot42^\circ\cot66^\circ\cot78^\circ = 1\)

  9. \(\tan12^\circ\tan24^\circ\tan48^\circ\tan84^\circ = 1\)

  10. \(\sin6^\circ\sin42^\circ\sin66^\circ\sin78^\circ = \frac{1}{16}\)

  11. \(\sin\frac{\pi}{5}\sin\frac{2\pi}{5}\sin\frac{3\pi}{5}\sin\frac{4\pi}{5} = \frac{5}{16}\)

  12. \(\cos36^\circ\cos72^\circ\cos108^\circ\cos144^\circ = \frac{1}{16}\)

  13. \(\cos\frac{\pi}{15}\cos\frac{2\pi}{15}\cos\frac{3\pi}{15}\cos\frac{4\pi}{15}\cos\frac{5\pi}{15}\cos\frac{6\pi}{15}\cos\frac{7\pi}{15} = \frac{1}{2^7}\)

  14. \(\cos\frac{\pi}{65}\cos\frac{2\pi}{65}\cos\frac{4\pi}{65}\cos\frac{8\pi}{65}\cos\frac{16\pi}{65}\cos\frac{32\pi}{65} = \frac{1}{64}\)

  15. If \(\tan \frac{A}{2} = \sqrt{\frac{a - b}{a + b}}\tan \frac{B}{2},\) prove that, \(\cos A = \frac{a\cos B + b}{a + b\cos B}\)

  16. If \(\tan \frac{A}{2} \ = \sqrt{\frac{1 - e}{1 + e}}\tan\frac{B}{2},\) prove that \(\cos B = \frac{\cos A - e}{1 - e\cos A}\)

  17. If \(\sin A + \sin B = a\) and \(\cos A + \cos B = b,\) prove that \(\sin(A + B) = \frac{2ab}{a^2 + b^2}\)

  18. If \(\sin A + \sin B = a\) and \(\cos A + \cos B = b,\) prove that \(\cos(A - B) = \frac{1}{2}(a^2 + b^2 - 2)\)

  19. If \(A\) and \(B\) be two different roots of equation \(a\cos\theta + b\sin\theta = c,\) prove that

    1. \(\tan(A + B) = \frac{2ab}{a^2 - b^2}\)

    2. \(\cos(A + B) = \frac{a^2 - b^2}{a^2 + b^2}\)

  20. If \(\cos A + \cos B = \frac{1}{3}\) and \(\sin A + \sin B = \frac{1}{4},\) prove that \(\cos \frac{A - B}{2} = \pm\frac{5}{24}\)

  21. If \(2\tan \frac{A}{2} = \tan \frac{B}{2},\) prove that \(\cos A = \frac{3 + 5\cos B}{5 + 3\cos B}\)

  22. If \(\sin A = \frac{4}{5}\) and \(\cos B = \frac{5}{13},\) prove that one value of \(\cos \frac{A - B}{2} = \frac{8}{\sqrt{65}}\)

  23. If \(\sec(A + B) + \sec(A - B) = 2\sec A,\) prove that \(\cos B = \pm \sqrt{2}\cos \frac{B}{2}\)

  24. If \(\cos \theta = \frac{\cos\alpha\cos\beta}{1 - \sin\alpha\sin\beta},\) prove that one of the values of \(\tan \frac{\theta}{2}\) is \(\frac{\tan \frac{\alpha}{2} - \tan\frac{\beta}{2}}{1 - \tan\frac{\alpha}{2}\tan\frac{\beta}{2}}\)

  25. If \(\tan\alpha = \frac{\sin\theta\sin\phi}{\cos\theta + \cos\phi},\) prove that one of the values of \(\tan\frac{\alpha}{2}\) is \(\tan\frac{\theta}{2}\tan\frac{\phi}{2}\)

  26. If \(\cos\theta = \frac{\cos\alpha + \cos\beta}{1 + \cos\alpha\cos\beta},\) prove that one of the values of \(\tan\frac{\theta}{2}\) is \(\tan\frac{\alpha}{2}\tan\frac{\beta}{2}\)