Question

5.
sin“ A-cost A=1-2 cos’ A.
Solution:​

Answer 1

Given to prove :-

• sin⁴ A - cos⁴ A = 1 - 2 cos² A

Answer :-

We have,

L.H.S = sin⁴ A - cos⁴ A

R.H.S = 1 - 2 cos² A

Therefore,

L.H.S = sin⁴ A - cos⁴ A

= (sin² A)² - (cos² A)²

= (sin² A + cos² A) (sin² A - cos² A)

By using the identity (sin² A + cos² A) = 1, we get,

= 1 (sin² A - cos² A)

By multiplying 1 with (sin² A - cos² A), we get,

= sin² A - cos² A

By using sin² A = (1 - cos² A), we get,

= (1 - cos² A) - cos² A

= 1 - 2 cos² A = R.H.S

Hence, Proved!

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Remember:-

• sin² A = (1 - cos² A)
• (sin² A + cos² A) = 1