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QJEEM22Maf4fb: JEE Main 2022 Mathematics - Class 12 JEEM22Maf4fb Solution: Considering the principal values of the inverse trigonometric…

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation cos⁡−1(x)−2sin⁡−1(x)=cos⁡−1(2x)\cos ^{-1}(x)-2 \sin ^{-1}(x)=\cos ^{-1}(2 x) is equal to :

Multiple Choice Options

  • A 0
  • B 1
  • C 12\frac{1}{2}
  • D −12-\frac{1}{2}

Step-by-Step Verified Solution

cos⁡−1x−2sin⁡−1x=cos⁡−12x{\cos ^{ - 1}}x - 2{\sin ^{ - 1}}x = {\cos ^{ - 1}}2x

For Domain : x∈[−12,12]x \in \left[ {{{ - 1} \over 2},{1 \over 2}} \right]

cos⁡−1x−2(π2−cos⁡−1x)=cos⁡−1(2x){\cos ^{ - 1}}x - 2\left( {{\pi \over 2} - {{\cos }^{ - 1}}x} \right) = {\cos ^{ - 1}}(2x)

⇒cos⁡−1x+2cos⁡−1x=π+cos⁡−12x\Rightarrow {\cos ^{ - 1}}x + 2{\cos ^{ - 1}}x = \pi + {\cos ^{ - 1}}2x

⇒cos⁡(3cos⁡−1x)=−cos⁡(cos⁡−12x)\Rightarrow \cos (3{\cos ^{ - 1}}x) = - \cos ({\cos ^{ - 1}}2x)

⇒4x3=x\Rightarrow 4{x^3} = x

⇒x=3, ± 12\Rightarrow x = 3,\, \pm \,{1 \over 2}

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