2sin^4x+3sin^2xcos^2x +cos^4x=1+sin^2x

Question

Verify each identity. show each change made.

Answer 1

Rewrite the left side as 2sin²xsin²x + 3sin²xcos²x + cos²xcos²x

= 2sin²x(1-cos²x) + 3sin²xcos²x + cos²xcos²x

= 2sin²x + sin²xcos²x + cos²xcos²x = 2sin²x + cos²x(sin²x + cos²x)

= 2sin²x + cos²x = 2sin²x + (1 - sin²x) = sin²x + 1

Answer 2

2 sin⁴ x + 3 sin² x cos ² x + cos⁴ x

(2 sin² x + 1)(sin² x + cos² x)

(2 sin² x + 1)( 1)

2 sin² + 1

QED