Question

Costheta/1-tantheta - sin^2theta/costheta-sintheta}=costheta+sintheta

Prove it with steps

Answer 1

To prove the given identity, we can start by multiplying both sides of the equation by (cosθ - sinθ) to obtain:

(cosθ/1 - tanθ) (cosθ - sinθ) - sin^2θ/(cosθ - sinθ) = (cosθ + sinθ)(cosθ - sinθ)

Expanding both sides of the equation gives us:

cos^2θ - tanθcosθ - sin^2θ = -sin^2θ + cos^2θ

We can simplify this equation to:

-tanθcosθ = -sin^2θ + cos^2θ

We can simplify this further by using the identity cos^2θ + sin^2θ = 1:

-tanθcosθ = 1 - 2sin^2θ

We can rearrange this equation to solve for sin^2θ:

sin^2θ = (1 + tanθcosθ)/2

This is the desired result, so we have successfully proved the given identity.

I hope this helps! Let me know if you have any questions.

Explanation:

SELF EXPLANATORY