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प्रश्न
Prove that:
`cot^(-1) ((sqrt(1+sin x) + sqrt(1-sinx))/(sqrt(1+sin x) - sqrt(1- sinx))) = x/2, x in (0, pi/4)`
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उत्तर
L.H.S. = `cot^(-1) ((sqrt(1+sin x) + sqrt(1-sinx))/(sqrt(1+sin x) - sqrt(1- sinx)))`
= `cot^(-1) (sqrt(1+sin x) + sqrt(1-sinx))/(sqrt(1+sin x) - sqrt(1 - sin x)) xx (sqrt(1+sin x) + sqrt(1-sinx))/(sqrt(1+sin x) - sqrt(1 - sin x))`
= `cot^(-1) ((1+sinx) + (1-sinx) + 2sqrt(1 - sin^2 x))/((1+sinx) - (1 - sinx)`
= `cot^(-1) (2(1 + cos x))/(2sin x)`
= `cot^(-1) (1+ cosx)/sin x`
= `cot^(-1) (2 cos^2 x/2)/(2sin x/2 cos x/2)`
= `cot^-1 (cot x/2)`
= `x/2`
L.H.S. = R.H.S.
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