KCET · Maths · Inverse Trigonometric Functions
If \( 3 \tan ^{-1} x+\cot ^{-1}=\pi \) then \( x \) equal to
- A \( 00 \)
- B \( \pi\)
- C \( -1 \)
- D \( \frac{1}{2} \)
Answer & Solution
Correct Answer
(B) \( \pi\)
Step-by-step Solution
Detailed explanation
Given that, \(3 \tan ^{-1} x+\cot ^{-1} x=\pi\)
\(\Rightarrow 2 \tan ^{-1} x+\left(\tan ^{-1} x+\cot ^{-1} x\right)=\pi\)
Since, \(\tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2} .\) Then
\(\Rightarrow 2 \tan ^{-1} x+\frac{\pi}{2}=\pi\)
\(\Rightarrow 2 \tan ^{-1} x=\frac{\pi}{2}\)
\(\Rightarrow \tan ^{-1} x=\frac{\pi}{4}\)
\(\Rightarrow x=\tan \frac{\pi}{4}=1\)
\(\Rightarrow 2 \tan ^{-1} x+\left(\tan ^{-1} x+\cot ^{-1} x\right)=\pi\)
Since, \(\tan ^{-1} x+\cot ^{-1} x=\frac{\pi}{2} .\) Then
\(\Rightarrow 2 \tan ^{-1} x+\frac{\pi}{2}=\pi\)
\(\Rightarrow 2 \tan ^{-1} x=\frac{\pi}{2}\)
\(\Rightarrow \tan ^{-1} x=\frac{\pi}{4}\)
\(\Rightarrow x=\tan \frac{\pi}{4}=1\)
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