Question

If sin theta + cos theta = root 2, then the value of sin⁵theta + cos⁵theta is ? The answer is (1)/2root2 . please show the steps ..​

Answer 1

EXPLANATION.

⇒ sinθ + cosθ = √2.

As we know that,

Squaring on both sides of the equation, we get.

⇒ (sinθ + cosθ)² = (√2)².

As we know that,

Formula of :

⇒ (x + y)² = x² + y² + 2xy.

⇒ sin²θ + cos²θ = 1.

⇒ (x² + y²) = (x + y)² - 2xy.

⇒ (x³ + y³) = (x + y)(x² - xy + y²).

Using this formula in the equation, we get.

⇒ sin²θ + cos²θ + 2sinθcosθ = 2.

⇒ 1 + 2sinθcosθ = 2.

⇒ 2sinθcosθ = 2 - 1.

⇒ 2sinθcosθ = 1.

⇒ sinθcosθ = 1/2.

To find : sin⁵θ + cos⁵θ.

⇒ sin⁵θ + cos⁵θ = (sin²θ + cos²θ)(sin³θ + cos³θ) - sin²θcos³θ - cos²θsin³θ.

⇒ sin⁵θ + cos⁵θ = (1)[(sinθ + cosθ)(sin²θ - sinθcosθ + cos²θ)] - sin²θcos²θ(cosθ + sinθ).

⇒ sin⁵θ + cos⁵θ = (1)(sinθ + cosθ)[(sinθ + cosθ)² - 3sinθcosθ] - (sinθcosθ)²(cosθ + sinθ).

Put the values in the equation, we get.

⇒ sin⁵θ + cos⁵θ = (1)(√2)[(√2)² - 3(1/2)] - (1/2)²(√2).

⇒ sin⁵θ + cos⁵θ = (√2)[2 - 3/2] - (√2/4).

⇒ sin⁵θ + cos⁵θ = (√2)[(4 - 3)/2] - (√2/4).

⇒ sin⁵θ + cos⁵θ = (√2)(1/2) - (√2/4).

⇒ sin⁵θ + cos⁵θ = (√2/2) - (√2/4).

⇒ sin⁵θ + cos⁵θ = (2√2 - √2)/4.

⇒ sin⁵θ + cos⁵θ = (√2)/4.

⇒ sin⁵θ + cos⁵θ = (√2)/4 x (√2)/(√2).

⇒ sin⁵θ + cos⁵θ = (2/4√2).

⇒ sin⁵θ + cos⁵θ = (1/2√2).