Prove that: `(2-cosec^2x) / (cosec^2x + 2cotx) = (sinx - cosx) / (sinx + cosx)`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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It has to be proved that: `(2 - cosec^2x)/(cosec^2 x + 2*cot x) = (sin x - cos x) / (sin x + cos x)`

`(2 - cosec^2x)/(cosec^2 x + 2*cot x)`

=> `(2 - 1/(sin^2x))/(1/(sin^2x) + (2*cosx)/(sin x))`

=> `((2*sin^2x)/(sin^2x) - 1/(sin^2x))/(1/(sin^2x) + (2*cosx*sin x)/(sin^2x))`

=> `(2*sin^2x - 1)/(1 + 2*cosx*sin x)` ...(1)

`(sin x - cos x) / (sin x + cos x)`

=> `((sin x - cos x)(sin x + cos x)) / ((sin x + cos x)(sin x + cos x))`

=> `(sin^2x - cos^2x)/(sin^2x + cos^2x + 2*sinx*cosx)`

=> `(sin^2x - 1 + sin^2x)/(1 + 2*sinx*cosx)`

=> `(2*sin^2x - 1)/(1 + 2*sinx*cosx)` ...(2)

(1) and (2) are the same.

This proves the identity `(2 - cosec^2x)/(cosec^2 x + 2*cot x) = (sin x - cos x) / (sin x + cos x)`

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