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BITSAT 2024Paper 1

Exam: BITSAT
Year: 2024
Paper: Paper 1
Subject: Mathematics
Chapter: Inverse Trigonometric Functions

BITSAT 2024 · Paper 1 · Mathematics · Inverse Trigonometric Functions

If ^ -1 ((1)/(2))= ( ^ -1 x) , then the value of x is

If

\cos \cot ^{-1} \left(\frac{1}{2}\right)=\cot\left(\cos ^{-1} x\right)

, then the value of

x is

Options

A $\frac{1}{\sqrt{6}}$
B $\frac{-1}{\sqrt{12}}$
C $\frac{2}{\sqrt{6}}$
D $\frac{-2}{\sqrt{6}}$

Correct answer

A. $\frac{1}{\sqrt{6}}$

Step-by-step solution

We have

\cos \left(\cot ^{-1}

\left(\frac{1}{2}\right)\right)=\cot \left(\cos ^{-1}

x\right)$ Let $\cot $^{-1}\left(\frac{1}{2}\right)=\alpha \Rightarrow \cot \alpha=\frac{1}{2} \Rightarrow \cos ^{-1} \frac{1}{\sqrt{5}}\Rightarrow \cos \left(\cos ^{-1} \frac{1}{\sqrt{5}}\right)=\cot \left(\cot ^{-1} \frac{x}{\sqrt{1-x^2}}\right)\Rightarrow \frac{1}{\sqrt{5}}=\frac{x}{\sqrt{1-x^2}} \Rightarrow \sqrt{1-x^2}=\sqrt{5}

x

On squaring both sides, we get,

1-x²=5 x² \Rightarrow 1=6 x²\Rightarrow x=

\pm

\frac{1}{\sqrt{6}}

\begin{array}{ll}& x=\frac{1}{\sqrt{6}} \quad \text{ [neglecting - ve sign] }\end{array}

$

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