Find the value of \( \cos\left(\sec^{-1}x + \csc^{-1}x\right), \quad |x| \ge 1 \)

Solution:

Let

\[ \theta = \sec^{-1}x \Rightarrow \sec \theta = x \Rightarrow \cos \theta = \frac{1}{x} \]

\[ \phi = \csc^{-1}x \Rightarrow \csc \phi = x \Rightarrow \sin \phi = \frac{1}{x} \]

Now,

\[ \sin \theta = \sqrt{1 – \frac{1}{x^2}} = \frac{\sqrt{x^2 – 1}}{x} \]

\[ \cos \phi = \sqrt{1 – \frac{1}{x^2}} = \frac{\sqrt{x^2 – 1}}{x} \]

Using identity:

\[ \cos(\theta + \phi) = \cos\theta \cos\phi – \sin\theta \sin\phi \]

\[ = \frac{1}{x} \cdot \frac{\sqrt{x^2 – 1}}{x} – \frac{\sqrt{x^2 – 1}}{x} \cdot \frac{1}{x} \]

\[ = 0 \]

Hence,

\[ \cos\left(\sec^{-1}x + \csc^{-1}x\right) = 0 \]

Final Answer:

\[ \cos\left(\sec^{-1}x + \csc^{-1}x\right) = 0 \]