The value of \(\cot \left(\cos^{-1}\displaystyle\frac{7}{25}\right)\) is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(\displaystyle\frac{7}{24}\)
Let
\( \displaystyle \theta = \cos^{-1}\frac{7}{25} \)
Then \(\cos\theta = \frac{7}{25} \gt 0\), so \(\theta\) lies in \(\left(0, \frac{\pi}{2}\right)\) (the principal value range of \(\cos^{-1}\) is \([0, \pi]\)) and \(\sin\theta \gt 0\):
\( \displaystyle \sin\theta = \sqrt{1 - \frac{49}{625}} \)
\( \displaystyle = \sqrt{\frac{576}{625}} = \frac{24}{25} \)
\( \displaystyle \cot\theta = \frac{\cos\theta}{\sin\theta} = \frac{7/25}{24/25} = \frac{7}{24} \)