cos( pi/3 + sec inverse x + cosec inverse x) = - root 3/2 then value of x is

Given the equation:

\[
\cos\left(\frac{\pi}{3} + \sec^{-1} x + \csc^{-1} x\right) = -\frac{\sqrt{3}}{2}
\]

We need to find the value of \(x\).

### Step 1: Simplify the expression

First, let's recognize the given cosine value:

\[
\cos\left(\frac{\pi}{3} + \sec^{-1} x + \csc^{-1} x\right) = -\frac{\sqrt{3}}{2}
\]

The cosine value \(-\frac{\sqrt{3}}{2}\) corresponds to an angle of \(\frac{5\pi}{6}\) (or \(\frac{7\pi}{6}\)).

### Step 2: Consider the identity for inverse trigonometric functions

For any \(x > 1\), we know:

\[
\sec^{-1} x = \theta \quad \text{implies} \quad \sec \theta = x
\]

\[
\csc^{-1} x = \phi \quad \text{implies} \quad \csc \phi = x
\]

So, \( \theta + \phi = \sec^{-1} x + \csc^{-1} x \).

### Step 3: Using sum of angles in trigonometric identities

Given that \(\cos\left(\frac{\pi}{3} + \theta + \phi\right) = -\frac{\sqrt{3}}{2}\), let's assume:

\[
\frac{\pi}{3} + \sec^{-1} x + \csc^{-1} x = \frac{5\pi}{6} \quad \text{or} \quad \frac{7\pi}{6}
\]

#### Case 1: \( \frac{\pi}{3} + \sec^{-1} x + \csc^{-1} x = \frac{5\pi}{6} \)

\[
\sec^{-1} x + \csc^{-1} x = \frac{5\pi}{6} - \frac{\pi}{3} = \frac{\pi}{2}
\]

However, for these inverse trigonometric functions, a specific identity often used is:

\[
\sec^{-1} x + \csc^{-1} x = \frac{\pi}{2} \quad \text{if} \quad x > \sqrt{2}
\]

This leads to:

\[
\frac{\pi}{3} + \frac{\pi}{2} = \frac{5\pi}{6}
\]

Thus, \(x\) must satisfy \(x = \sqrt{2}\).

#### Case 2: \( \frac{\pi}{3} + \sec^{-1} x + \csc^{-1} x = \frac{7\pi}{6} \)

This case does not lead to a simple solution since it results in inconsistent identities for the angles within standard principal ranges.

### Conclusion

The value of \(x\) that satisfies the given equation is:

\[
\boxed{\sqrt{2}}
\]

---

Would you like a more detailed explanation or have any other questions? Here are 5 related questions that might interest you:

1. What are the properties of inverse trigonometric functions like \(\sec^{-1} x\) and \(\csc^{-1} x\)?
2. How do trigonometric identities help in solving inverse trigonometric equations?
3. What is the range and domain of \(\sec^{-1} x\) and \(\csc^{-1} x\)?
4. How can the sum of angles identity be applied to more complex trigonometric equations?
5. How do principal values affect solutions in trigonometric equations?

**Tip:** When dealing with inverse trigonometric functions, remember that certain identities and constraints on the domain are crucial for finding valid solutions.

Learn how to solve the trigonometric equation cos(pi/3 + sec^-1 x + csc^-1 x) = -sqrt(3)/2 step-by-step. Understand the identity involving sum of angles in trigonometry and properties of inverse trigonometric functions. Suitable for advanced high school students.

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