Question

prove that cos²72°-sin²54=-√5/4​

Answer 1

Answer:

ANSWER

cos²72 = cos²(90 - 18) = sin²18

sin²54 = sin²(90 - 36) = cos²36

Firstly find out the value of sin(18°) using the compound angle formulae as shown.

Answer 2

Step-by-step explanation:

Let us consider the LHS

sin2 42° – cos2 78° = sin2 (90° – 48°) – cos2 (90° – 12°)

= cos2 48° – sin2 12° [since, sin (90 – A) = cos A and cos (90 – A) = sin A]

As we know, cos (A + B) cos (A – B) = cos2A – sin2B

Now the above equation becomes,

= cos2 (48° + 12°) cos (48° – 12°)

= cos 60° cos 36° [since, cos 36° = (√5 + 1)/4]

= 1/2 × (√5 + 1)/4

= (√5 + 1)/8

= RHS

Thus proved.