\frac{1+\sin ^2\left(x\right)}{\cos ^2\left(x\right)}=1+2\tan ^2\left(x\right)

To verify the identity:

\[
\frac{1 + \sin^2(x)}{\cos^2(x)} = 1 + 2\tan^2(x)
\]

we will manipulate the left-hand side (LHS) and right-hand side (RHS) to see if they are indeed equal.

### Step 1: Express the LHS
Starting with the left-hand side:

\[
\frac{1 + \sin^2(x)}{\cos^2(x)}
\]

We can split the fraction:

\[
\frac{1}{\cos^2(x)} + \frac{\sin^2(x)}{\cos^2(x)}
\]

This can be rewritten as:

\[
\sec^2(x) + \tan^2(x)
\]

### Step 2: Express the RHS
Now, let's simplify the right-hand side:

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

### Step 3: Compare LHS and RHS
We need to check if:

\[
\sec^2(x) + \tan^2(x) \stackrel{?}{=} 1 + 2\tan^2(x)
\]

Using the Pythagorean identity:

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

Substitute this into the LHS:

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

This simplifies to:

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

Thus, the LHS equals the RHS, verifying the identity:

\[
\frac{1 + \sin^2(x)}{\cos^2(x)} = 1 + 2\tan^2(x)
\]

This identity is correct.

Would you like any further details or clarifications?

Here are 5 related questions:
1. How can the Pythagorean identities be derived from the unit circle?
2. What is the significance of the tangent function in trigonometry?
3. Can you prove the identity \(\sec^2(x) = 1 + \tan^2(x)\)?
4. How would this identity change if we used \(\sin(x)\) and \(\cos(x)\) in different combinations?
5. What other identities involve \(\sec(x)\) and \(\tan(x)\)?

**Tip:** Always check trigonometric identities by breaking them down using fundamental identities like the Pythagorean identities or by simplifying both sides separately.

Learn how to verify the trigonometric identity (1 + sin^2(x)) / cos^2(x) = 1 + 2 tan^2(x) with a detailed step-by-step solution. This problem involves manipulating trigonometric expressions using the Pythagorean identity and the tangent identity. Suitable for advanced high school students.

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