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**5. If \( \sin \theta = \cos \theta \), find the value of \( 2 \tan^2 \theta + 3 \sin^2 \theta - 1 \).**

Let's solve this step-by-step.

### Step 1: Solving \( \sin \theta = \cos \theta \)

We know that \( \sin \theta = \cos \theta \) implies:

\[
\frac{\sin \theta}{\cos \theta} = 1
\]

This simplifies to:

\[
\tan \theta = 1
\]

Thus, \( \theta = 45^\circ \) or \( \theta = \frac{\pi}{4} \) radians.

### Step 2: Calculate \( 2 \tan^2 \theta + 3 \sin^2 \theta - 1 \)

Since \( \tan \theta = 1 \), we can substitute into the expression:

\[
\tan^2 \theta = 1^2 = 1
\]
\[
\sin \theta = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}
\]
\[
\sin^2 \theta = \left( \frac{\sqrt{2}}{2} \right)^2 = \frac{2}{4} = \frac{1}{2}
\]

Now, substitute these values into the expression:

\[
2 \tan^2 \theta + 3 \sin^2 \theta - 1 = 2(1) + 3\left( \frac{1}{2} \right) - 1
\]
\[
= 2 + \frac{3}{2} - 1
\]
\[
= 2 - 1 + \frac{3}{2} = 1 + \frac{3}{2} = \frac{2}{2} + \frac{3}{2} = \frac{5}{2}
\]

### Final Answer:
The value of the expression is \( \frac{5}{2} \).

Would you like any more details or explanations? Here are some related questions you might consider:

1. What is the general solution to \( \sin \theta = \cos \theta \)?
2. How does the identity \( \tan^2 \theta + 1 = \sec^2 \theta \) relate to this problem?
3. What are the exact values of trigonometric functions for other angles like \( 30^\circ \) or \( 60^\circ \)?
4. How do you convert between radians and degrees?
5. What is the double-angle formula for sine, and how is it derived?

**Tip**: Always remember that \( \sin \theta = \cos \theta \) occurs when \( \theta = 45^\circ + n \cdot 180^\circ \) where \( n \) is an integer, considering the periodicity of trigonometric functions.

If sin θ = cos θ, find the value of 2tan²θ + 3sin²θ - 1.

Step-by-step solution to the trigonometric problem where sin θ = cos θ. Learn how to calculate the value of 2tan²θ + 3sin²θ - 1 by using key trigonometric identities, suitable for students in Grades 9-12.

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