The given equation is:

\[
\sec^2x + \tan^x = 7
\]

I believe you meant \(\sec^2x + \tan^2x = 7\), because the secant-tangent identity in trigonometry involves the square of \(\tan x\), not just \(\tan x\). This identity is given by:

\[
\sec^2x = 1 + \tan^2x
\]

Using this identity, we can substitute \(\sec^2x\) into the equation:

\[
(1 + \tan^2x) + \tan^2x = 7
\]

Simplifying the equation:

\[
1 + 2\tan^2x = 7
\]

Subtract 1 from both sides:

\[
2\tan^2x = 6
\]

Now divide both sides by 2:

\[
\tan^2x = 3
\]

Taking the square root of both sides:

\[
\tan x = \pm \sqrt{3}
\]

So, \(x\) satisfies:

\[
x = \tan^{-1}(\sqrt{3}) \quad \text{or} \quad x = \tan^{-1}(-\sqrt{3})
\]

The solutions for \(x\) will be:

\[
x = \frac{\pi}{3} + n\pi \quad \text{or} \quad x = \frac{2\pi}{3} + n\pi \quad \text{for any integer } n
\]

Would you like more details on the steps? Let me know!

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Here are five related questions:

1. What is the tangent-secant identity in trigonometry?
2. How do you solve for angles when \(\tan x = \sqrt{3}\)?
3. How does the periodicity of the tangent function affect the solutions?
4. What are the other basic trigonometric identities and their importance?
5. How do we find general solutions for trigonometric equations?

**Tip:** Always check if a trigonometric equation uses standard identities to simplify the solving process.

Explore how to solve the trigonometric equation sec^2x + tan^2x = 7 using the secant-tangent identity. Follow the step-by-step solution, including key trigonometric identities and the general solutions for x. Ideal for students in Grades 10-12.

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