Express each of the following a sum or difference of two trigonometric

1.	Express each of the following a sum or difference of two trigonometric functions. (i) sin 5A · cos3A (ii) cos 4A sin 2A (iii) cos \(\frac{5\theta}{2}\). sin\(\frac{\theta}{2}\) (iv) 2 cos 70° cos 10°
Answer» (i) Using transformation formulae sin5A . cos3A = \(\frac{1}{2}\)[sin(5A +3A) + sin(5A – 3A)] = \(\frac{1}{2}\)(sin8A + sin2A) (ii) cos4A – sin2A = \(\frac{1}{2}\)[sin 6A – sin2A] (iii) cos \(\frac{5\theta}{2}\). sin\(\frac{\theta}{2}\) = \(\frac{1}{2}\)[sin3θ – sin2θ] (iv) 2cos70° . cos10° = 2 x \(\frac{1}{2}\)[cos80° + cos60°] = cos80° +\(\frac{1}{2}\)

Express each of the following a sum or difference of two trigonometric functions. (i) sin 5A · cos3A (ii) cos 4A sin 2A (iii) cos \(\frac{5\theta}{2}\). sin\(\frac{\theta}{2}\) (iv) 2 cos 70° cos 10°

Answer»

(i) Using transformation formulae

sin5A . cos3A = \(\frac{1}{2}\)[sin(5A +3A) + sin(5A – 3A)]

= \(\frac{1}{2}\)(sin8A + sin2A)

(ii) cos4A – sin2A = \(\frac{1}{2}\)[sin 6A – sin2A]

(iii) cos \(\frac{5\theta}{2}\). sin\(\frac{\theta}{2}\) = \(\frac{1}{2}\)[sin3θ – sin2θ]

(iv) 2cos70° . cos10° = 2 x \(\frac{1}{2}\)[cos80° + cos60°]

= cos80° +\(\frac{1}{2}\)