Question

If sinθ+sin²θ = 1, what is the value of cos²θ+cos⁴ θ?
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Answer 1

Answer:

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Given ,

sinθ + sin²θ = 1

To find ,

The value of : cos²θ + cos⁴θ

Now ,

sinθ + sin²θ = 1

sinθ = 1 - sin²θ

sinθ = cos²θ ---------- ( i )

[ • As sin²θ + cos²θ = 1

So , sin²θ = 1 - cos²θ ]

★ Method - 1

sinθ = cos²θ

( sinθ )² = ( cos²θ )²

sin²θ = cos⁴θ

1 - cos²θ = cos⁴θ

cos⁴θ + cos²θ = 1 [ ★ Required answer ]

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★ Method - 2

cos²θ + cos⁴θ

= sinθ + ( sinθ )²

[ • Putting the value of cos²θ = sinθ ]

= sinθ + sin²θ

= 1 [ • Given , sinθ + sin²θ = 1 ]

• So finally ,

[ cos²θ + cos⁴θ = 1 ]

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OR :

From equation (1)

sinθ + sin²θ=1

sin²θ=1-sinθ

(we know that sin²θ=1-cos²θ)

1-cos²θ=1-sinθ

cos²θ=sinθ

cos⁴θ =sin²θ

Now

cos⁴θ =sin²θ

cos⁴θ=1-cos²θ

cos²θ+cos⁴θ=1✔

Hope it helps

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Answer 2

Step-by-step explanation:

$\sin( \alpha ) + { \sin( \alpha ) }^{2} = 1$

$\sin( \alpha ) = 1 - { \sin( \alpha ) }^{2}$

$\sin( \alpha ) = { \cos( \alpha ) }^{2}$

so cos^2(a)+cos^4(a)= sin a+ sin^2 a

which is 1