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Question
If y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`
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Solution
y = `log[sqrt((1 - cos((3x)/2))/(1 +cos((3x)/2)))]`, find `("d"y)/("d"x)`
= `log[sqrt((2sin^2 ((3x)/4))/(2cos^2 ((3x)/4)))]`
= `log[sqrt(tan^2((3x)/4))]`
= `log[tan((3x)/4)]`
Differentiating w. r. t. x, we get
`("d"y)/("d"x) = "d"/("d"x)[log(tan((3x)/4))]`
= `1/(tan((3x)/4))* "d"/"d"x[tan((3x)/4)]`
= `cot((3x)/4)*sec^2((3x)/4)*"d"/("d"x)((3x)/4)`
= `cos((3x)/4)/(sin((3x)/4))*1/(cos^2((3x)/4))*3/4`
= `3/(2[2sin((3x)/4)cos((3x)/4)]`
= `3/(2sin((3x)/2))`
= `3/2"cosec"((3x)/2)`
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