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If-4-sin-1-x-cos-1-x-pi-then-x-equals-

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Question Number 113824 by deepraj123 last updated on 15/Sep/20
If 4 sin^(−1) x+cos^(−1) x= π, then x equals
If4sin−1x+cos−1x=π,thenxequals
Answered by $@y@m last updated on 15/Sep/20
Let sin^(−1) x=α ⇒sin α=x ...(1) Let cos ^(−1) x=β ⇒ cos β=x ...(2) ATQ, 4α+β=π 4α=π−β cos 4α=cos (π−β) cos 4α=−cos β cos 4α=−sin α 1−sin^2 2α=−sin α sin^2 2α=1+sin α (2sin αcos α)^2 =1+sin α (2x(√(1−x^2 )))^2 =1+x 4x^2 (1−x^2 )=1+x 4x^2 (1+x)(1−x)=1+x (1+x)(4x^2 −4x^3 −1)=0 x=−1
Letsin−1x=α⇒sinα=x…(1)Letcos−1x=β⇒cosβ=x…(2)ATQ,4α+β=π4α=π−βcos4α=cos(π−β)cos4α=−cosβcos4α=−sinα1−sin22α=−sinαsin22α=1+sinα(2sinαcosα)2=1+sinα(2x1−x2)2=1+x4x2(1−x2)=1+x4x2(1+x)(1−x)=1+x(1+x)(4x2−4x3−1)=0x=−1

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