Question

prove that cos4A=8cos⁴A-Acos²A+1+1​

Answer 1

L.H.S. = cos 4A

= cos 2(2A)

= 1 – 2 sin2 2A

= 1 – 2(2 sin A cos A)2

= 1 – 8 sin2A cos2A

= 1 – 8 (1 – cos2A) cos2A [1 – cos2A = sin2A]

= 1 – 8 cos2A + 8cos4A

= R.H.S.

HOPE THIS HELPS YOU

Answer 2

Required Answer:-

Given to Prove:

• cos(4x) = 8cos⁴(x) - 8cos²(x) + 1

Proof:

Taking LHS,

We know that,

➡ cos(2x) = 2cos²(x) - 1

Therefore,

cos(4x)

= 2cos²(2x) - 1

= 2{cos(2x)}² - 1

= 2{2cos²(x) - 1}² - 1

= 2{4cos⁴(x) + 1 - 4cos²(x)} - 1

= 8cos⁴(x) + 2 - 8cos²(x) - 1

= 8cos⁴(x) - 8cos²(x) + 2 - 1

= 8cos⁴(x) - 8cos²(x) + 1

= RHS (Hence Proved)

Formula Used:

• cos(2x) = 2cos²(x) - 1