Question

Prove that- sin^4a + cos^4a = 1- 2sin^2a.cos^2a

Answer 1

open the identity (sin²A+cos²A)²
as (a+b)²=a²+b²+2ab
(sin²A+cos²A)²=sin^4A+cos^4A+2sin²Acos²A        use sin²a+cos²a=1
(1)² - 2sin²Acos²A = sin^4A+cos^4A
1-2sin²Acos²A = sin^4A+cos^4A
hence proved

Answer 2

To Prove: sin⁴A + cos⁴A = 1 - 2sin²A × cos²A

Solution: sin⁴A + cos⁴A can be expressed as;

α² + β² = (α + β)² - 2αβ

(sin²A)² + (cos²A)² = (sin²A + cos²A)² - 2(sin²A)(cos²A)

$\boxed{\sf sin^{2}\theta + cos^{2}\theta = 1}$

(sin²A)² + (cos²A)² = (1)² - 2(sin²A)(cos²A)

(sin²A)² + (cos²A)² = 1 - 2 × sin²A × cos²A

Hence Proved.

Identities used in the Solution:

α² + β² = (α + β)² - 2αβ

sin²θ + cos²θ = 1