Question

Prove that

sin^2 24° - sin^2 6° = √5 - 1 / 8​

Answer 1

Answer:

Step-by-step explanation:

sin²24° - sin²6° = sin(24°+6°) sin(24°-6°)= sin 30°sin 18°

= $\frac{1}{2}$ sin 18°

let, 18°= x

sin(2x)= sin(90°-3x)=cos 3x

► 2sinx cosx= 4cos³x- 3cosx

► 2sinx = 4cos²x - 3= 4 - 4sin²x -3

► 4sin²x +2sinx -1 =0

► sinx = $\frac{-2 + \sqrt{4+16} }{8}$

►sinx = $\frac{-1 + \sqrt{5} }{4}$

$\frac{1}{2}$ sin 18° =  $\frac{1}{2}$ $\frac{-1 + \sqrt{5} }{4}$ = $\frac{ \sqrt{5} -1}{8}$