Question

If cosec theta + cot theta=k
then p.t. cos theta =k²-1/k²+1​

Answer 1

Answer:

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Answer 2

Answer:

Step-by-step explanation:

Given:

⇒cosecθ+cotθ=k --------(1),

By the trignometry identity , cosec^2A-cot^2A=1,

⇒cosec^2θ-cot^2θ=1,

⇒(cosecθ+cotθ)(cosecθ-cotθ)=1,

⇒(cosecθ-cotθ)*k=1,

⇒(cosecθ-cotθ)=1/k ----------(2),

By equation1+equation2 we get:

⇒2cosecθ=k+1/k=k^2+1/k,

⇒cosecθ=k^2+1/2k,

⇒1/sinθ=k^2+1/2k,

⇒sinθ=2k/k^2+1,

Now,we know that,cosA=√(1-sin^2A),

⇒cosA=√1-(2k/k^2+1)^2,

⇒cosA=√1-4k^2/(k^2+1)^2,

⇒√(k^2+1)^2-4k^2/(k^2+1)^2,

⇒√k^4+2k^2+1-4k^2/(k^2+1)^2,

⇒√(k^2-1)^2/(k^2+1)^2,

⇒k^2-1/k^2+1​,

So cos theta =k^2-1/k^2+1​,proved.

Hope it helps you.