InterviewSolution
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InterviewSolution
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सिद्ध कीजिए - `cot^(-1)[(sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sin x)-sqrt(1-sinx))]=(x)/(2), x in (0,(pi)/(4)).` |
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Answer» `L.H.S."=[(sqrt(1+sinx)+sqrt(1-sinx))/(sqrt(1+sinx)-sqrt(1-sinx))]` `=cot^(-1)[(sqrt(cos^(2).(x)/(2)+sin^(2).(x)/(2)+2sin.(x)/(2)cos.(x)/(2))+sqrt(cos^(2).(x)/(2)+sin^(2).(x)/(2)-2sin.(x)/(2)cos.(x)/(2)))/(sqrt(cos^(2).(x)/(2)+sin^(2).(x)/(2)+2sin.(x)/(2)cos.(x)/(2))-sqrt(cos^(2).(x)/(2)+sin^(2).(x)/(2)-2sin.(x)/(2)cos.(x)/(2)))],` `[because cos^(2).(x)/(2)+sin^(2).(x)/(2)-1" और "sinx=2sin.(x)/(2)cos.(x)/(2)]` `=cot^(-1)[(sqrt((cos.(x)/(2)+sin.(x)/(2))^(2))+sqrt((cos.(x)/(2)-sin.(x)/(2))))/(sqrt((cos.(x)/(2)+sin.(x)/(2))^(2))-sqrt((cos.(x)/(2)-sin.(x)/(2))^(2)))]` `=cot^(-1)[((cos.(x)/(2)+sin.(x)/(2))+(cos.(x)/(2)-sin.(x)/(2)))/((cos.(x)/(2)-sin.(x)/(2))-(cos.(x)/(2)-sin.(x)/(2)))],` `[because 0lt (x)/(2)lt(pi)/(4)" और "cos.(x)/(2) gt 0, sin.(x)/(2) gt 0]` `=cot^(-1)[(2cos.(x)/(2))/(2sin.(x)/(2))]` `=cot^(-1)(cot.(x)/(2))` `=(x)/(2)` = R.H.S.`" "`यही सिद्ध करना था। |
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