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प्रश्न
Simplest form of `tan^-1 ((sqrt(1 + cos "x") + sqrt(1 - cos "x"))/(sqrt(1 + cos "x") - sqrt(1 - cos "x")))`, `π < "x" < (3π)/2` is:
पर्याय
`π/4 - "x"/2`
`(3π)/2 - "x"/2`
`-"x"/2`
`π - "x"/2`
MCQ
उत्तर
`π/4 - "x"/2`
Explanation:
`tan^-1 ((sqrt(1 + cos "x") + sqrt(1 - cos "x"))/(sqrt(1 + cos "x") - sqrt(1 - cos "x")))`
= `tan^-1 ((-sqrt2 cos "x"/2 + sqrt2 sin "x"/2)/(-sqrt2 cos "x"/2 - sqrt2 sin "x"/2))`, `π < "x" < (3π)/2 ⇒ π/2 < "x"/2 < (3π)/4`
= `tan^-1 ((cos "x"/2 - sin "x"/2)/(cos "x"/2 + sin "x"/2))`
= `tan^-1 ((1 - tan "x"/2)/(1 + tan "x"/2))`
= `tan^-1 [tan(π/4 - "x"/2)]`
= `π/4 - "x"/2, -π/4 > π/4 - "x"/2 > -π/2`
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