Question

2 sin² A + cos⁴ A – 1 is equal to ______.

Options

• sin⁴ A

• cos² A

• tan² A

• tan⁴ A

Answer 1

2 sin² A + cos⁴ A – 1 is equal to sin⁴ A.

Explanation:

sin² A + cos² A = 1

cos² A = 1 − sin² A

2 sin² A + (1 − sin² A)² − 1

Use the algebraic identity (a − b)² = a² − 2ab + b² where a = 1 and b = sin² A:

(1 − sin² A)² = 1 − 2 sin² A + (sin² A)²

(1 − sin² A)² = 1 − 2 sin² A + sin⁴ A

2 sin² A + (1 − 2 sin² A + sin⁴ A) − 1

(2 sin² A − 2 sin² A) + (1 − 1) + sin⁴ A

0 + 0 + sin⁴ A = sin⁴ A