Solve for x: \sin ^{-1}(1-x)-2 \sin ^{-1}(x)=\frac{\pi}{2}.

CBSE Class 12 Math 2020 Delhi Set 1 Solved Paper - Question 27

Section - C

Q. Nos. 27 to 32 carry 4 marks each.

Question : 27 of 36

Marks: +1, -0

Solve for

x: sin \;^{-1}(1-x)-2 sin \;^{-1}(x)=\;{π}/{2}

Solution:

Given that

sin \;^{-1}(1-x)-2 sin \;^{-1} x=\;{π}/{2}

let

{\x}= sin \; {\y}

∴ sin \;^{-1}(1- sin \; {\y})-2 {\y}=\;{π}/{2}

⇒ sin \;^{-1}(1- sin \; {\y})=\;{π}/{2}+2 {\y}

⇒ 1- sin \; {\y}= sin \; (\;{π}/{2}+2 {\y})

⇒ 1- sin \; {\y}=\cos 2 {\y}

⇒ 1- sin \; {\y}=1-2 sin \;^2 {\y} \;\text " as "\; \cos 2 {\y}=1-2 sin \;^2 {\y}

⇒ 2 sin \;^2 {\y}- sin \; {\y}=0

⇒ 2 {\x}^2- {\x}=0

⇒ {\x}(2 {\x}-1)=0

⇒ {\x}=0,2 {\x}-1=0

⇒ {\x}=0,2 {\x}=1

⇒ {\x}=0, {\x}=\;{1}/{2}

But

{\x}=\;{1}/{2}

does not satisfy the given equation

∴ {\x}=0

is the solution of the given equation.

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