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Question

Question 1
If cosec θ+cot θ=p, then prove that cos θ=p21p2+1.

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Solution

Given, cosec θ+cot θ=p

1sin θ+cos θsin θ=p[cosec θ=1sin θ and cot θ=cos θsin θ]

1+cos θsin θ=p1

(1+cos θ)2sin2 θ=p21 [take square on both sides]

1+cos2 θ+2 cos θsin2 θ=p21

Using componendo and dividendo rule, we get;

(1+cos2 θ+2 cos θ)sin2 θ(1+cos2 θ+2 cos θ)+sin2 θ=p21p2+1

1+cos2 θ+2 cos θ(1cos2 θ)1+2 cos θ+(cos2 θ+sin2 θ)=p21p2+1 [sin2 θ+cos2 θ=1]

2 cos2 θ+2 cos θ2+2 cos θ=p21p2+1

2 cos θ(cos θ+1)2(cos θ+1)=p21p2+1

cos θ=p21p2+1

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