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Given: cos A+cos² A = 1

(Rearranging, and taking cos² A on the right side of equation)

⇒ cos A = 1- cos² A

(∵ sin² θ+cos² θ = 1 ⇒ sin² θ = 1- cos² θ)

⇒ cos A = sin² A …eq. 1

Squaring both sides,

we get cos² A = sin⁴ A …eq. 2

We have to find sin²A+sin⁴ A=1

So, adding eq. 1 and eq. 2, we get

sin²A + sin⁴ A= cos A + cos² A (As given)

∴ sin²A+ sin⁴ A = 1