Question

$sec\left(\cos e{c}^{-1}x\right)$ is equal to

• A. $\cos ec\left(se{c}^{-1}x\right)$
• B. $cotx$
• C. $\mathrm{\pi }$
• D. None of these

Answer 1

The correct option is A

$\cos ec\left(se{c}^{-1}x\right)$

Explanation for the correct option:

Finding trigonometric relations to solve the given equation:

Assume the value of ${cosec}^{-1}x=A$

Then, the value of $cosecA=x$ and hence $\sin A=\frac{1}{x}$.

Using the trigonometric identity to get the cosine value.

$$\begin{gathered} \Rightarrow {\sin }^{2}A+{\cos }^{2}A=1 \\ \Rightarrow \cos A=\sqrt{1-{\sin }^{2}A} \\ \Rightarrow \cos A=\sqrt{1-\frac{1}{x^2}} \\ \Rightarrow \cos A=\frac{\sqrt{x^2-1}}{x} \end{gathered}$$

So, using the obtained value to find the secant value.

$$\begin{gathered} \Rightarrow \sec A=\frac{x}{\sqrt{x^2-1}}\left[\because cosx=\frac{1}{secx}\right] \\ \Rightarrow \text{sec}\left(\cos e{c}^{-1}x\right)=\frac{x}{\sqrt{x^2-1}}\mathbf{[}\mathbf{\because }\mathit{c}\mathit{o}\mathit{s}\mathit{e}{\mathit{c}}^{\mathbf{-}\mathbf{1}}\mathit{x}\mathbf{=}\mathit{A}\mathbf{,}\mathbf{}\mathit{a}\mathit{s}\mathit{s}\mathit{u}\mathit{m}\mathit{e}\mathit{d}] \end{gathered}$$

Again, assume ${\text{sec}}^{-1}x=B$

Then, the value of $\sec B=x$

So, $\cos B=\frac{1}{x}$

Determining the sine value for the assumption.

Use the trigonometric identity to get the sine value.

$\begin{array}{rcl}{\sin }^{2}B+{\cos }^{2}B& =& 1\\ & \Rightarrow & \sin B=\sqrt{1-{\cos }^{2}B}\\ & =& \sqrt{1-\frac{1}{x^2}}\\ & =& \frac{\sqrt{x^2-1}}{x}\end{array}$

So, using the obtained value to find the cosecant value.

$\begin{array}{rcl}\text{cosec}B& =& \frac{x}{\sqrt{x^2-1}}\\ & \Rightarrow & \text{cosec}\left({\text{sec}}^{-1}x\right)=\frac{x}{\sqrt{x^2-1}}\end{array}$

So, $\sec \left({cosec}^{-1}x\right)=cosec\left({\sec }^{-1}x\right)$

Hence, option (A) is the correct answer.