Question

if sinθ + sin²θ = 1, Prove that cos²θ + cos⁴θ = 1

Please give answer with solution.

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Answer 1

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Answer 2

$\huge\underline\mathfrak\red{Question-}$

★ If sinθ + sin²θ = 1, Prove that cos²θ + cos⁴θ = 1

$\huge\underline\mathfrak\red{Solution-}$

Given :

• sinθ + sin²θ = 1

To prove :

• cos²θ + cos⁴θ = 1

Proof :

It is given that, sinθ + sin²θ = 1

$\implies$ sinθ = 1 - sin²θ

$\implies$ sinθ = 1 - sin²θ

$\implies$ sinθ = cos²θ

Squaring on both sides :

( sinθ )² = ( cos²θ )²

$\implies$ ( sin²θ ) = cos⁴θ

$\implies$ ( 1 - cos²θ ) = cos⁴θ

$\implies$ 1 - cos²θ = cos⁴θ

$\implies$ 1 = cos⁴θ + cos²θ

$\implies$ cos⁴θ + cos²θ = 1

Hence proved!

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$\underline\bold\green{Trignometrical\:identity\:used-}$

• Sin²θ + cos²θ = 1