Question

sin theta\1+cos theta + 1+cos theta\sin theta= 2cosec theta

Answer 1

2/sin theta=2cosec theta

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Answer 2

Solution:(Instead of θ, I use A)

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Given:

To prove : $\frac{sinA}{1 + cosA} + \frac{1+cosA}{sinA} = 2 cosec A$

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Proof:

LHS = $\frac{sinA}{1+cosA} + \frac{1+cosA}{sinA}$

=> $\frac{(sinA)^2 + (1+cosA)^2}{sinA(1+cosA)}$

=> $\frac{sin^2A + 1+2cosA+cosA^2}{sinA(1+cosA)}$

=> $\frac{sin^2A +cosA^2+ 1+2cosA}{sinA(1+cosA)}$

We know that,

sin²A + cos²A = 1,

So,

=> $\frac{1+ 1+2cosA}{sinA(1+cosA)}$

=> $\frac{2+2cosA}{sinA(1+cosA)}$

=> $\frac{2(1+cosA)}{sinA(1+cosA)}$

=> $\frac{2}{sinA}$

=> $2( \frac{1}{sinA} )$

We know that ,

$\frac{1}{sinA} = cosecA$

So,

=> 2(cosec A)

=> 2cosec A,

=> RHS,

                 ∴ Hence proved,.

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