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# Prove that: sin^{6} x + cos^{6} x = 1 - 3 sin^{2} x cos^{2} x.

Cosine and Sine are primary trigonometric functions. They are also known as complemetary functions

## Answer: sin^{6} x + cos^{6} x = 1 - 3 sin^{2} x cos^{2} x

To solve this, we will use algebraic identity.

**Explanation:**

sin^{6} x + cos^{6} x can be written as (sin^{2}x )^{3} + (cos^{2}x )^{3}

Using Algebraic identity,

a^{3} + b^{3} = (a + b)^{3 }- 3ab (a + b)

(sin^{2}x )^{3} + (cos^{2}x )^{3} = (sin^{2}x + cos^{2}x )^{3 } - 3 sin^{2}x cos^{2}x (sin^{2}x + cos^{2}x )

Using trigonometric identity sin^{2}x + cos^{2}x = 1, we get

⇒ sin^{6} x + cos^{6} x = (1) ^{3 }- 3 sin^{2}x cos^{2}x (1)

⇒ sin^{6} x + cos^{6} x = 1^{ }- 3 sin^{2}x cos^{2}x

### Thus, proved that sin^{6} x + cos^{6} x = 1 - 3 sin^{2} x cos^{2} x

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