MHT CET · Maths · Inverse Trigonometric Functions
The value of \(\tan \left(2 \tan ^{-1}\left(\frac{\sqrt{5}-1}{2}\right)\right)\) is
- A \(2 \sqrt{5}\)
- B 4
- C 2
- D \(\sqrt{5}-1\)
Answer & Solution
Correct Answer
(C) 2
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \tan \left(2 \tan ^{-1}\left(\frac{\sqrt{5}-1}{2}\right)\right) \\ & =\frac{2 \tan \left(\tan ^{-1}\left(\frac{\sqrt{5}-1}{2}\right)\right)}{1-\tan ^2\left(\tan ^{-1}\left(\frac{\sqrt{5}-1}{2}\right)\right)} \\ & =\frac{2\left(\frac{\sqrt{5}-1}{2}\right)}{1-\left(\frac{\sqrt{5}-1}{2}\right)^2} \\ & =\frac{\sqrt{5}-1}{1-\left(\frac{6-2 \sqrt{5}}{4}\right)} \\ & =\frac{4(\sqrt{5}-1)}{2 \sqrt{5}-2} \\ & =\frac{4(\sqrt{5}-1)}{2(\sqrt{5}-1)}=2\end{aligned}\)
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