10. Transformation Formulae#

10.1. Transformation of products into sums or differences#

We know that sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin(A + B) = \sin A\cos B + \cos A\sin B

and sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin(A - B) = \sin A\cos B - \cos A\sin B

Adding these, we get 2sin⁡Acos⁡B=sin⁡(A+B)+sin⁡(A−B)2\sin A\cos B = \sin(A + B) + \sin(A - B)

Subtracting, we get 2cos⁡Asin⁡B=sin⁡(A+B)−sin⁡(A−B)2\cos A\sin B = \sin(A + B) - \sin (A - B)

We also know that cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos (A + B) = \cos A\cos B - \sin A\sin B

and cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos(A - B) = \cos A\cos B + \sin A\sin B

Adding, we get 2cos⁡Acos⁡B=cos⁡(A+B)+cos⁡(A−B)2\cos A\cos B = \cos (A + B) + \cos(A - B)

Subtrating we get 2sin⁡sin⁡B=cos⁡(A−B)−cos⁡(A+B)2\sin \sin B = \cos (A - B) - \cos(A + B)

10.2. Transformation of sums or differences into products#

We have 2sin⁡Acos⁡B=sin⁡(A+B)sin⁡(A−B)2\sin A\cos B = \sin(A + B)\sin(A - B)

Substituting for A+B=C,A−B=DA + B = C, A - B = D so that A=C+D2A = \frac{C + D}{2} and B=C−D2B = \frac{C- D}{2}

sin⁡C+sin⁡D=2sin⁡C+D2cos⁡C−D2\sin C + \sin D = 2\sin \frac{C + D}{2}\cos \frac{C - D}{2}

We also have 2cos⁡Asin⁡B=sin⁡(A+B)−sin⁡(A−B)2\cos A\sin B = \sin(A + B) - \sin (A - B)

Following similarly sin⁡C−sin⁡D=2cos⁡C+D2sin⁡C−D2\sin C - \sin D = 2\cos \frac{C + D}{2}\sin \frac{C - D}{2}

For 2cos⁡Acos⁡B=cos⁡(A+B)+cos⁡(A−B)2\cos A\cos B = \cos (A + B) + \cos(A - B)

we get cos⁡C+cos⁡D=2cos⁡C+D2cos⁡C−D2\cos C + \cos D = 2\cos \frac{C + D}{2}\cos \frac{C - D}{2}

For 2sin⁡sin⁡B=cos⁡(A−B)−cos⁡(A+B)2\sin \sin B = \cos (A - B) - \cos(A + B)

we get cos⁡C−cos⁡D=2sin⁡C+D2sin⁡D−C2\cos C - \cos D = 2\sin \frac{C + D}{2}\sin \frac{D - C}{2}

10.3. Problems#

  1. Find the value of sin⁡75∘−sin⁡15∘cos⁡75∘+cos⁡15∘\frac{\sin 75^\circ - \sin 15^\circ}{\cos 75^\circ + \cos 15^\circ}

  2. Simplify the expression (cos⁡θ−cos⁡3θ)(sin⁡8θ+sin⁡2θ)(sin⁡5θ−sin⁡θ)(cos⁡4θ−cos⁡6θ)\frac{(\cos \theta - \cos 3\theta)(\sin 8\theta + \sin 2\theta)}{(\sin 5\theta - \sin\theta)(\cos 4\theta - \cos 6\theta)}

Prove that

  1. sin⁡7θ−sin⁡5θcos⁡7θ+cos⁡5θ=tan⁡θ\frac{\sin7\theta - \sin5\theta}{\cos7\theta + \cos5\theta} = \tan\theta

  2. cos⁡6θ−cos⁡4θsin⁡6θ+sin⁡4θ=−tan⁡θ\frac{\cos6\theta - \cos4\theta}{\sin6\theta + \sin4\theta} = -\tan\theta

  3. sin⁡A+sin⁡3Acos⁡A+cos⁡3A=tan⁡2A\frac{\sin A + \sin 3A}{\cos A + \cos 3A} = \tan 2A

  4. sin⁡7A−sin⁡Asin⁡8A−sin⁡2A=cos⁡4Asec⁡5A\frac{\sin 7A - \sin A}{\sin 8A - \sin 2A} = \cos 4A\sec 5A

  5. cos⁡2B+cos⁡2Acos⁡2B−cos⁡2A=cot⁡(A+B)cot⁡(A−B)\frac{\cos 2B + \cos 2A}{\cos 2B - \cos 2A} = \cot(A + B)\cot(A - B)

  6. sin⁡2A+sin⁡2Bsin⁡2A−sin⁡2B=tan⁡(A+B)tan⁡(A−B)\frac{\sin 2A + \sin 2B}{\sin 2A - \sin 2B} = \frac{\tan(A + B)}{\tan(A - B)}

  7. sin⁡A+sin⁡2Acos⁡A−cos⁡2A=cot⁡A2\frac{\sin A + \sin 2A}{\cos A - \cos 2A} = \cot \frac{A}{2}

  8. sin⁡5A−sin⁡3Acos⁡3A+cos⁡5A=tan⁡A\frac{\sin 5A - \sin 3A}{\cos 3A + \cos 5A} = \tan A

  9. cos⁡2B−cos⁡2Asin⁡2B+sin⁡2A=tan⁡(A−B)\frac{\cos 2B - \cos 2A}{\sin 2B + \sin 2A} = \tan(A - B)

  10. cos⁡(A+B)+sin⁡(A−B)=2sin⁡(45∘+A)cos⁡(45∘+B)\cos (A + B) + \sin(A - B) = 2\sin(45^\circ + A)\cos(45^\circ + B)

  11. cos⁡3A−cos⁡Asin⁡3A−sin⁡A+cos⁡2A−cos⁡4Asin⁡4A−sin⁡2A=sin⁡Acos⁡2Acos⁡3A\frac{\cos 3A - \cos A}{\sin 3A - \sin A} + \frac{\cos 2A - \cos 4A}{\sin 4A - \sin 2A} = \frac{\sin A}{\cos 2A\cos 3A}

  12. sin⁡(4A−2B)+sin⁡(4B−2A)cos⁡(4A−2B)+cos⁡(4B−2A)=tan⁡(A+B)\frac{\sin (4A - 2B) + \sin (4B - 2A)}{\cos (4A - 2B) + \cos (4B - 2A)} = \tan(A + B)

  13. tan⁡5θ+tan⁡3θtan⁡5θ−tan⁡3θ=4cos⁡2θcos⁡4θ\frac{\tan 5\theta + \tan 3\theta}{\tan 5\theta - \tan 3\theta} = 4\cos 2\theta\cos 4\theta

  14. cos⁡3θ+2cos⁡5θ+cos⁡7θcos⁡θ+2cos⁡3θ+cos⁡5θ=cos⁡2θ−sin⁡2θtan⁡3θ\frac{\cos 3\theta + 2\cos5\theta + \cos 7\theta}{\cos\theta + 2\cos3\theta + \cos 5\theta} = \cos 2\theta - \sin 2\theta\tan 3\theta

  15. sin⁡A+sin⁡3A+sin⁡5A+sin⁡7Acos⁡A+cos⁡3A+cos⁡5A+cos⁡7A=tan⁡4A\frac{\sin A + \sin 3A + \sin 5A + \sin 7A}{\cos A + \cos 3A + \cos 5A + \cos 7A} = \tan 4A

  16. sin⁡(θ+ϕ)−2sin⁡θ+sin⁡(θ−ϕ)cos⁡(θ+ϕ)−2cos⁡θ+cos⁡(θ−ϕ)=tan⁡θ\frac{\sin (\theta + \phi) - 2\sin\theta + \sin (\theta - \phi)}{\cos (\theta + \phi) - 2\cos \theta + \cos(\theta - \phi)} = \tan\theta

  17. sin⁡A+2sin⁡3A+sin⁡5Asin⁡3A+2sin⁡5A+sin⁡7A=sin⁡3Asin⁡5A\frac{\sin A + 2\sin 3A + \sin 5A}{\sin 3A + 2\sin 5A + \sin 7A} = \frac{\sin 3A}{\sin 5A}

  18. sin⁡(A−C)+2sin⁡A+sin⁡(A+C)sin⁡(B−C)+2sin⁡B+sin⁡(B+C)=sin⁡Asin⁡B\frac{\sin(A - C) + 2\sin A + \sin(A + C)}{\sin (B - C) + 2\sin B + \sin(B + C)} = \frac{\sin A}{\sin B}

  19. sin⁡A−sin⁡5A+sin⁡9A−sin⁡13Acos⁡A−cos⁡5A−cos⁡9A+cos⁡13A=cot⁡4A\frac{\sin A - \sin 5A + \sin 9A - \sin 13A}{\cos A - \cos 5A - \cos 9A + \cos 13 A} = \cot 4A

  20. sin⁡A+sin⁡Bsin⁡A−sin⁡B=tan⁡A+B2cot⁡A−B2\frac{\sin A + \sin B}{\sin A - \sin B} = \tan \frac{A + B}{2}\cot \frac{A - B}{2}

  21. cos⁡A+cos⁡Bcos⁡B−cos⁡A=cot⁡A+B2cot⁡A−B2\frac{\cos A + \cos B}{\cos B - \cos A} = \cot \frac{A + B}{2}\cot \frac{A - B}{2}

  22. sin⁡A+sin⁡Bcos⁡A+cos⁡B=tan⁡A+B2\frac{\sin A + \sin B}{\cos A + \cos B} = \tan \frac{A + B}{2}

  23. sin⁡A−sin⁡Bcos⁡B−cos⁡A=cot⁡A+B2\frac{\sin A - \sin B}{\cos B - \cos A} = \cot \frac{A + B}{2}

  24. cos⁡(A+B+C)+cos⁡(−A+B+C)+cos⁡(A−B+C)+cos⁡(A+B−C)sin⁡(A+B+C)+sin⁡(−A+B+C)−sin⁡(A−B+C)+sin⁡(A+B−C)=cot⁡B\frac{\cos(A + B + C) + \cos(-A + B + C) + \cos(A - B + C) + \cos(A + B - C)}{\sin(A + B + C)+\sin(-A + B + C) - \sin(A - B + C) + \sin(A + B - C)} = \cot B

  25. cos⁡3A+cos⁡5A+cos⁡7A+cos⁡15A=4cos⁡4Acos⁡5Acos⁡6A\cos 3A + \cos 5A + \cos 7A + \cos 15A = 4 \cos 4A\cos 5A \cos 6A

  26. cos⁡(−A+B+C)+cos⁡(A−B+C)+cos⁡(A+B−C)+cos⁡(A+B+C)=4cos⁡Acos⁡Bcos⁡C\cos(-A + B + C) + \cos(A - B + C) + \cos(A + B - C) + \cos(A + B + C) = 4\cos A\cos B\cos C

  27. sin⁡50∘−sin⁡70∘+sin⁡10∘=0\sin 50^\circ - \sin 70^\circ + \sin 10^\circ = 0

  28. sin⁡10∘+sin⁡20∘+sin⁡40∘+sin⁡50∘=sin⁡70∘+sin⁡80∘\sin 10^\circ + \sin 20^\circ + \sin 40^\circ + \sin 50^\circ = \sin 70^\circ + \sin 80^\circ

  29. sin⁡α+sin⁡2α+sin⁡4α+sin⁡5α=4cos⁡α2cos⁡3α2sin⁡3α\sin\alpha + \sin 2\alpha + \sin 4\alpha + \sin 5\alpha = 4\cos \frac{\alpha}{2}\cos \frac{3\alpha}{2}\sin 3\alpha

Simplify:

  1. cos⁡[θ+(n−32)ϕ]−cos⁡[θ+(n+32)ϕ]\cos\left[\theta + \left(n - \frac{3}{2}\right)\phi\right] - \cos\left[\theta + \left(n + \frac{3}{2}\right)\phi\right]

  2. sin⁡[θ+(n−32)ϕ]+sin⁡[θ+(n+32)ϕ]\sin\left[\theta + \left(n - \frac{3}{2}\right)\phi\right] + \sin\left[\theta + \left(n + \frac{3}{2}\right)\phi\right]

Express as a sum or difference the following:

  1. 2sin⁡5θsin⁡7θ2\sin5\theta\sin7\theta

  2. 2cos⁡7θsin⁡5θ2\cos7\theta\sin5\theta

  3. 2cos⁡11θcos⁡3θ2\cos 11\theta\cos 3\theta

  4. 2sin⁡54∘sin⁡66∘2\sin54^\circ\sin66^\circ

Prove that

  1. sin⁡θ2sin⁡7θ2+sin⁡3θ2sin⁡11θ2=sin⁡2θsin⁡5θ\sin\frac{\theta}{2}\sin\frac{7\theta}{2} + \sin \frac{3\theta}{2}\sin\frac{11\theta}{2} =\sin 2\theta\sin 5\theta

  2. cos⁡2θcos⁡θ2−cos⁡3θcos⁡9θ2=sin⁡5θsin⁡5θ2\cos 2\theta\cos \frac{\theta}{2} -\cos3\theta\cos\frac{9\theta}{2} = \sin5\theta\sin\frac{5\theta}{2}

  3. sin⁡Asin⁡(A+2B)−sin⁡Bsin⁡(B+2A)=sin⁡(A−B)sin⁡(A+B)\sin A\sin(A + 2B) - \sin B\sin(B + 2A) = \sin(A - B)\sin(A + B)

  4. (sin⁡3A+sin⁡A)sin⁡A+(cos⁡3A−cos⁡A)cos⁡A=0(\sin 3A + \sin A)\sin A + (\cos 3A - \cos A)\cos A = 0

  5. 2sin⁡(A−C)cos⁡C−sin⁡(A−2C)2sin⁡(B−C)cos⁡C−sin⁡(B−2C)=sin⁡Asin⁡B\frac{2\sin(A - C)\cos C - \sin(A - 2C)}{2\sin(B - C)\cos C - \sin(B - 2C)} = \frac{\sin A}{\sin B}

  6. sin⁡Asin⁡2A+sin⁡3Asin⁡6A+sin⁡4Asin⁡13Asin⁡Acos⁡2A+sin⁡3Acos⁡6A+sin⁡4Acos⁡13A=tan⁡9A\frac{\sin A\sin 2A + \sin 3A\sin 6A + \sin4A\sin 13A}{\sin A\cos2A + \sin 3A\cos 6A + \sin 4A\cos 13A} = \tan 9A

  7. cos⁡2Acos⁡3A−cos⁡2Acos⁡7A+cos⁡Acos⁡10Asin⁡4Asin⁡3A−sin⁡2Asin⁡5A+sin⁡4Asin⁡7A=cot⁡6Acot⁡5A\frac{\cos 2A\cos 3A - \cos 2A\cos 7A + \cos A\cos 10A}{\sin 4A\sin 3A - \sin 2A\sin 5A + \sin 4A\sin 7A} =\cot 6A\cot 5A

  8. cos⁡(36∘−A)cos⁡(36∘+A)+cos⁡(54∘+A)cos⁡(54∘−A)=cos⁡2A\cos(36^\circ - A)\cos(36^\circ + A) + \cos(54^\circ + A)\cos(54^\circ - A) = \cos 2A

  9. cos⁡Asin⁡(B−C)+cos⁡Bsin⁡(C−A)+cos⁡Csin⁡(A−B)=0\cos A\sin(B - C) + \cos B\sin(C - A) + \cos C\sin(A - B) = 0

  10. sin⁡(45∘+A)sin⁡(45∘−A)=12cos⁡2A\sin(45^\circ + A)\sin(45^\circ - A) = \frac{1}{2}\cos 2A

  11. sin⁡(β−γ)cos⁡(α−δ)+sin⁡(γ−α)cos⁡(β−δ)+sin⁡(α−β)cos⁡(γ−δ)=0\sin(\beta - \gamma)\cos(\alpha - \delta) + \sin(\gamma - \alpha)\cos(\beta - \delta) + \sin(\alpha - \beta)\cos(\gamma - \delta) = 0

  12. 2cos⁡π13cos⁡9π13+cos⁡3π13+cos⁡5π13=02\cos\frac{\pi}{13}\cos \frac{9\pi}{13} + \cos \frac{3\pi}{13} + \cos \frac{5\pi}{13} = 0

  13. cos⁡55∘+cos⁡65∘+cos⁡175∘=0\cos 55^\circ + \cos65^\circ + \cos 175^\circ = 0

  14. cos⁡18∘−sin⁡18∘=2sin⁡27∘\cos 18^\circ -\sin 18^\circ = \sqrt{2}\sin 27^\circ

  15. sin⁡A+sin⁡2A+sin⁡4A+sin⁡5Acos⁡A+cos⁡2A+cos⁡4A+cos⁡5A=tan⁡3A\frac{\sin A + \sin 2A + \sin 4A + \sin 5A}{\cos A + \cos 2A + \cos 4A + \cos 5A} = \tan 3A

  16. (cos⁡A+cos⁡Bsin⁡A−sin⁡A)n+(sin⁡A+sin⁡Bcos⁡A−cos⁡B)n=2cot⁡nA−B2\left(\frac{\cos A + \cos B}{\sin A - \sin A}\right)^n + \left(\frac{\sin A + \sin B}{\cos A - \cos B}\right)^n = 2\cot^n \frac{A - B}{2} or 00 accordingh as nn is even or odd.

  17. If α,β,γ\alpha, \beta, \gamma are in A.P., show that cos⁡β=sin⁡α−sin⁡γcos⁡γ−cos⁡α\cos\beta = \frac{\sin\alpha - \sin\gamma}{\cos\gamma - \cos\alpha}

  18. If sin⁡θ+sin⁡ϕ=3(cos⁡ϕ−cos⁡θ)\sin\theta + \sin\phi = \sqrt{3}(\cos\phi - \cos\theta) prove that sin⁡3θ+sin⁡3ϕ=0\sin3\theta + \sin3\phi = 0

  19. sin⁡65∘+cos65∘=2cos⁡20∘\sin 65^\circ + cos 65^\circ = \sqrt{2}\cos 20^\circ

  20. sin⁡47∘+cos⁡77∘=cos⁡17∘\sin 47^\circ + \cos 77^\circ = \cos 17^\circ

  21. cos⁡10∘−sin⁡10∘cos⁡10∘+sin⁡10∘=tan⁡35∘\frac{\cos 10^\circ - \sin 10^\circ}{\cos 10^\circ + \sin 10^\circ} = \tan 35^\circ

  22. cos⁡80∘+cos⁡40∘−cos20∘=0\cos 80^\circ + \cos 40^\circ - cos 20^\circ = 0

  23. cos⁡π5+cos⁡2π5+cos⁡6π5+cos⁡7π5=0\cos\frac{\pi}{5} + \cos \frac{2\pi}{5} + \cos\frac{6\pi}{5} + \cos \frac{7\pi}{5} = 0

  24. cos⁡α+cos⁡β+cos⁡γ+cos⁡(α+β+γ)=4cos⁡α+β2cos⁡β+γ2cos⁡γ+α2\cos\alpha + \cos\beta + \cos\gamma + \cos(\alpha + \beta + \gamma) = 4\cos\frac{\alpha + \beta}{2}\cos\frac{\beta + \gamma}{2}\cos \frac{\gamma + \alpha}{2}

  25. If sin⁡α−sin⁡β=13\sin\alpha - \sin\beta = \frac{1}{3} and cos⁡β−cos⁡α=12,\cos\beta - \cos\alpha = \frac{1}{2}, prove that cot⁡α+β2=23\cot\frac{\alpha + \beta}{2} = \frac{2}{3}

  26. If cosec⁡A+secA=cosec⁡B+sec⁡B,\cosec A + sec A = \cosec B + \sec B, prove that tan⁡Atan⁡B=cot⁡A+B2\tan A\tan B = \cot \frac{A + B}{2}

  27. If sec⁡(θ+α)+sec⁡(θ−α)=2sec⁡θ,\sec(\theta + \alpha) + \sec(\theta - \alpha) = 2\sec\theta, show that cos⁡2θ=1+cos⁡α\cos^2\theta = 1 + \cos\alpha

  28. Show that sin⁡50∘cos⁡85∘=1−2sin⁡35∘22\sin50^\circ\cos85^\circ = \frac{1 - \sqrt{2}\sin 35^\circ}{2\sqrt{2}}

  29. Prove that sin⁡20∘sin⁡40∘sin⁡80∘=38\sin 20^\circ \sin 40^\circ\sin 80^\circ = \frac{\sqrt{3}}{8}

  30. Prove that sin⁡Asin⁡(60∘−A)sin⁡(60∘+A)=14sin⁡3A\sin A\sin(60^\circ - A)\sin(60^\circ + A) = \frac{1}{4}\sin 3A

  31. If α+β=90∘,\alpha + \beta = 90^\circ, find the maximum value of sin⁡αsin⁡β\sin\alpha\sin\beta

  32. Prove that sin⁡25∘cos⁡115∘=12(sin⁡40∘−1)\sin 25^\circ\cos 115^\circ = \frac{1}{2}(\sin 40^\circ - 1)

  33. Prove that sin⁡20∘sin⁡40∘sin⁡60∘sin⁡80∘=316\sin 20^\circ \sin 40^\circ\sin 60^\circ \sin80^\circ = \frac{3}{16}

  34. Prove that cos⁡20∘cos⁡40∘cos⁡80∘=18\cos 20^\circ\cos40^\circ\cos80^\circ = \frac{1}{8}

  35. Prove that tan⁡20∘tan⁡40∘tan⁡60∘tan⁡80∘=3\tan20^\circ\tan40^\circ\tan60^\circ\tan80^\circ = 3

  36. Prove that cos⁡10∘cos⁡30∘cos⁡50∘cos⁡70∘=316\cos10^\circ\cos30^\circ\cos50^\circ\cos70^\circ = \frac{3}{16}

  37. Prove that 4cos⁡θcos⁡(π3+θ)cos⁡(π3−θ)=cos⁡3θ4\cos\theta\cos\left(\frac{\pi}{3} + \theta\right)\cos\left(\frac{\pi}{3} - \theta\right) = \cos3\theta

  38. Prove that tan⁡θtan⁡(60∘−θ)tan⁡(60∘+θ)=tan⁡3θ\tan\theta\tan(60^\circ - \theta)\tan(60^\circ + \theta) = \tan3\theta

  39. If α+β=90∘,\alpha + \beta = 90^\circ, show that the maximum value of cos⁡αcos⁡β\cos\alpha\cos\beta is 12\frac{1}{2}

  40. If cos⁡α=12,sin⁡β=13,\cos\alpha = \frac{1}{\sqrt{2}}, \sin\beta = \frac{1}{\sqrt{3}}, show that tan⁡α+β2cot⁡α−β2=5+26\tan\frac{\alpha + \beta}{2}\cot\frac{\alpha - \beta}{2} = 5 + 2\sqrt{6} or 5−265- 2\sqrt{6 }

  41. If xcos⁡θ=ycos⁡(θ+2π3)=zcos⁡(θ+4π3),x\cos\theta = y\cos\left(\theta + \frac{2\pi}{3}\right) = z\cos\left(\theta + \frac{4\pi}{3}\right), prove that xy+yz+xz=0xy + yz + xz = 0

  42. If sin⁡θ=nsin⁡(θ+2α),\sin\theta = n\sin(\theta + 2\alpha), prove that tan⁡(θ+α)=1+n1−ntan⁡α\tan(\theta + \alpha) = \frac{1 + n}{1 - n}\tan\alpha

  43. If sin⁡(θ+α)cos⁡(θ−α)=1−m1+m,\frac{\sin(\theta + \alpha)}{\cos(\theta - \alpha)} = \frac{1 - m}{1 + m}, prove that tan⁡(π4−θ)tan⁡(π4−α)=m\tan\left(\frac{\pi}{4} - \theta\right)\tan\left(\frac{\pi}{4} - \alpha\right) = m

  44. If ysin⁡ϕ=xsin⁡(2θ+ϕ),y\sin\phi = x\sin(2\theta + \phi), show that (x+y)cot⁡(θ+ϕ)=(y−x)cot⁡θ(x + y)\cot(\theta + \phi) = (y - x)\cot\theta

  45. If cos⁡(α+β)sin⁡(γ+δ)=cos⁡(α−beta)sin⁡(γ−δ),\cos(\alpha + \beta)\sin(\gamma + \delta) = \cos(\alpha - beta)\sin(\gamma - \delta), prove that cot⁡αcot⁡βcot⁡γ=cotδ\cot\alpha\cot\beta\cot\gamma = cot\delta

  46. If cos⁡(A−B)cos⁡(A+B)+cos⁡(C+D)cos⁡(C−D)=0,\frac{\cos(A - B)}{\cos(A + B)} + \frac{\cos(C + D)}{\cos(C - D)} = 0, prove that tan⁡Atan⁡Btan⁡Ctan⁡D=−1\tan A\tan B\tan C\tan D = -1

  47. If tan⁡(θ+ϕ)=3tan⁡θ,\tan(\theta + \phi) = 3\tan\theta, prove that sin⁡(2θ+ϕ)=2sin⁡ϕ\sin(2\theta + \phi) = 2\sin\phi

  48. If tan⁡(θ+ϕ)=3tan⁡θ,\tan(\theta + \phi) = 3\tan\theta, prove that sin⁡2(θ+ϕ)+sin⁡2θ=2sin⁡2ϕ\sin2(\theta + \phi) + \sin2\theta = 2\sin2\phi