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If y={(log)(cosx)sinx}{(log)(sinx)cosx}^...

If `y={(log)_(cosx)sinx}{(log)_(sinx)cosx}^(-1)+sin^(-1)((2x)/(1+x^2)),fin d(dy)/(dx)a tx=pi/4`

Text Solution

Verified by Experts

The correct Answer is:
`(-8)/(log e^(2))+(32)/(16+pi^(2)) `
Given, `y={(log_(cos x) sin x)*(log_(sin x) cos x)^(-1) + sin ((2x)/(1+x^(2)))}`
` :. " " y= ((log_(e)(sin x))/(log_(e)(cos x)))^(2) + sin ^(-1)((2x)/(1+x^(2)))`
` rArr (dy)/(dx) =2 {(log_(e)(sin x))/(log_(e)(cos x))*(log_(e)(cos x)*cot x + log_(e) ("in "x)*tan x)/({log_(e)(cos x)}^(2))}+2/(1+x^(2))`
`rArr ((dy)/(dx))_((x=pi/4))=2{1*(2*log(1/sqrt2))/((log. 1/sqrt2)^(2))}+2/(1+pi^(2)/16 )`
` = - 8/(log_(e) 2) +32/(16+pi^(2))`
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