https://file.aiquickdraw.com/mathbot/file/17253990610007ruyltxx.jpg

The equation to solve is:

\[
\tan^2 \alpha + 1 = \frac{1}{\cos^2 \alpha}
\]

To solve this, let's break it down step-by-step:

### Step 1: Use a Trigonometric Identity

Recall the Pythagorean identity for tangent and secant:

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

Since \(\sec \alpha = \frac{1}{\cos \alpha}\), we can express \(\sec^2 \alpha\) as:

\[
\sec^2 \alpha = \frac{1}{\cos^2 \alpha}
\]

### Step 2: Substitute the Identity

Now, substitute the identity into the given equation:

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

So,

\[
\tan^2 \alpha + 1 = \frac{1}{\cos^2 \alpha}
\]

### Step 3: Confirm the Equation Holds

Since both sides of the equation are equal, the given expression is an identity, meaning it is true for all values of \(\alpha\) where \(\cos \alpha \neq 0\) (because division by zero is undefined).

### Conclusion

The equation is verified to be correct based on the trigonometric identity, and no further solving is necessary. The equation is valid for all angles \(\alpha\) where \(\cos \alpha \neq 0\).

Would you like any more details or have any questions?

### Related Questions

1. What are other fundamental trigonometric identities?
2. How do we derive the Pythagorean identity for sine and cosine?
3. How do we use trigonometric identities to solve complex trigonometric equations?
4. Can we derive \(\tan \alpha\) in terms of \(\sin \alpha\) and \(\cos \alpha\)?
5. How does the unit circle help in understanding trigonometric functions?

### Tip

Always remember the fundamental trigonometric identities; they are key to simplifying and solving various trigonometric problems.

Solve the equation tan^2(α) + 1 = 1 / cos^2(α).

Learn how to solve the trigonometric equation tan^2(α) + 1 = 1/cos^2(α) using fundamental trigonometric identities. This problem covers key concepts like the Pythagorean Identity and is suitable for students in Grades 10-12.

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