Question

cos 10° cos 30° cos 50° cos 70°= 3/16 Prove that​

Answer 1

(cos70°·cos10°)·(cos 50°·cos 30°) $\frac{3}{16}$, it is proved.

Step-by-step explanation:

L.H.S. = (cos70°·cos10°)·(cos 50°·cos 30°)

Multiplying and dividing by 2, we get

= $\frac{1}{2}$·(2cos70°·cos10°))·cos 50°·cos 30°)

= $\frac{1}{2}$·(cos(70° + 10° ) + cos(70° - 10° ))·cos 50°·cos 30°)

= $\frac{1}{2}$·(cos80° + cos 60°)·cos 50°·cos 30°)

Multiplying and dividing by 2, we get

= $\frac{1}{4}$·(cos80° + cos 60°)·(2cos50°·cos 30°)

= $\frac{1}{4}$·(cos80° + cos 60°)·(2cos50°·$\frac{\sqrt{3} }{2}$)

= $\frac{\sqrt{3} }{8}$)·(cos80° + cos 60°)·(2cos50°)

= $\frac{\sqrt{3} }{8}$)·(2cos80°·cos 50°+ 2·cos 60°·cos 50°)

= $\frac{\sqrt{3} }{8}$)·(cos 130° + cos 30° + 2·$\frac{1}{2}$·cos 50°)

= $\frac{\sqrt{3} }{8}$)·(cos (180° - 50°) + $\frac{\sqrt{3} }{2}$ + cos 50°)

= $\frac{\sqrt{3} }{8}$)·(- cos 50° + $\frac{\sqrt{3} }{2}$ + cos 50°)

= $\frac{\sqrt{3} }{8}$)·( $\frac{\sqrt{3} }{2}$ )

= $\frac{3}{16}$

= R.H.S.

Hence, it is proved.