Question

Prove that following identity costheta /1-tan the tarsi square theta /cos theta -sintheta = costheta+sintheta​

Answer 1

The correct question be:

Prove that,

{cosθ / (1 - tanθ)} - {sin²θ / (cosθ - sinθ)} = cosθ + sinθ

Proof:

L.H.S. = {cosθ / (1 - tanθ)} - {sin²θ / (cosθ - sinθ)}

= {cosθ / (1 - sinθ/cosθ)} - {sin²θ / (cosθ - sinθ)}

= cos²θ/(cosθ - sinθ) - sin²θ/(cosθ - sinθ)

= (cos²θ - sin²θ) / (cosθ - sinθ)

= {(cosθ + sinθ) (cosθ - sinθ)} / (cosθ - sinθ)

= cosθ + sinθ = R.H.S.

Hence, proved.

Note to remember:

While solving trigonometric problems of this type where the right hand side contains sine or cosine ratios of any angle 'θ', be ready to express the cosec, sec, tan or cot ratios of that angle in terms of sine or cosine. You will find the solution easier.

Some rules:

tanθ = sinθ / cosθ

a² - b² = (a + b) (a - b)