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The image asks to find the values of \(\sin 75^\circ\) and \(\tan 15^\circ\).

Let's break these down.

### 1. Finding \(\sin 75^\circ\):
Using a known identity:
\[
\sin(75^\circ) = \sin(45^\circ + 30^\circ)
\]
The sum formula for sine is:
\[
\sin(A + B) = \sin A \cos B + \cos A \sin B
\]
Substituting \(A = 45^\circ\) and \(B = 30^\circ\):
\[
\sin(75^\circ) = \sin(45^\circ) \cos(30^\circ) + \cos(45^\circ) \sin(30^\circ)
\]
\[
\sin(75^\circ) = \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \times \frac{1}{2}
\]
\[
\sin(75^\circ) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}
\]

### 2. Finding \(\tan 15^\circ\):
Using the identity for \(\tan(A - B)\), we can express \(\tan 15^\circ\) as:
\[
\tan(15^\circ) = \tan(45^\circ - 30^\circ)
\]
The difference formula for tangent is:
\[
\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A \tan B}
\]
Substituting \(A = 45^\circ\) and \(B = 30^\circ\):
\[
\tan(15^\circ) = \frac{\tan(45^\circ) - \tan(30^\circ)}{1 + \tan(45^\circ) \tan(30^\circ)}
\]
\[
\tan(15^\circ) = \frac{1 - \frac{1}{\sqrt{3}}}{1 + 1 \times \frac{1}{\sqrt{3}}}
\]
\[
\tan(15^\circ) = \frac{\frac{\sqrt{3} - 1}{\sqrt{3}}}{\frac{\sqrt{3} + 1}{\sqrt{3}}} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}
\]

Would you like more detailed explanations or further clarifications on these calculations?

Here are some follow-up questions:
1. What are sum and difference identities for trigonometric functions?
2. How do you derive exact values for trigonometric ratios of special angles?
3. Can the tangent function ever be undefined, and when does that happen?
4. What is the significance of using exact values versus decimal approximations in trigonometry?
5. How does the unit circle help in understanding these trigonometric functions?

**Tip:** Understanding the sum and difference identities is essential for simplifying more complex trigonometric expressions efficiently.

Learn how to calculate sin 75° and tan 15° using trigonometric sum and difference identities. This step-by-step guide includes key formulas and concepts for students in high school trigonometry.

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