To show that $\tan 15^\circ + \cot 15^\circ = 4$, we will first express both $\tan 15^\circ$ and $\cot 15^\circ$ in terms of trigonometric identities and then simplify.

Step 1: Express $\tan 15^\circ$ in terms of known angles

We can use the angle subtraction identity for tangent: $\tan(45^\circ - 30^\circ) = \frac{\tan 45^\circ - \tan 30^\circ}{1 + \tan 45^\circ \cdot \tan 30^\circ}$

We know: $\tan 45^\circ = 1, \quad \tan 30^\circ = \frac{1}{\sqrt{3}}$

Substitute these values into the identity: $\tan 15^\circ = \frac{1 - \frac{1}{\sqrt{3}}}{1 + 1 \cdot \frac{1}{\sqrt{3}}}$

Simplify the expression: $\tan 15^\circ = \frac{\frac{\sqrt{3} - 1}{\sqrt{3}}}{\frac{\sqrt{3} + 1}{\sqrt{3}}} = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}$

Step 2: Express $\cot 15^\circ$

Since $\cot \theta = \frac{1}{\tan \theta}$, we have: $\cot 15^\circ = \frac{1}{\tan 15^\circ} = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}$

Step 3: Calculate $\tan 15^\circ + \cot 15^\circ$

Now add the two expressions: $\tan 15^\circ + \cot 15^\circ = \frac{\sqrt{3} - 1}{\sqrt{3} + 1} + \frac{\sqrt{3} + 1}{\sqrt{3} - 1}$

To add these fractions, find a common denominator: $\text{Common denominator} = (\sqrt{3} + 1)(\sqrt{3} - 1) = (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2$

Now, calculate the numerators: $\tan 15^\circ + \cot 15^\circ = \frac{(\sqrt{3} - 1)^2 + (\sqrt{3} + 1)^2}{2}$

Expand the squares in the numerator: $(\sqrt{3} - 1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3}$ $(\sqrt{3} + 1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3}$

Add these results: $4 - 2\sqrt{3} + 4 + 2\sqrt{3} = 8$

Thus: $\tan 15^\circ + \cot 15^\circ = \frac{8}{2} = 4$

Conclusion:

$\tan 15^\circ + \cot 15^\circ = 4$

This completes the proof.

Would you like more details on any step? Here are some related questions you might find helpful:

1. How do you derive the tangent addition and subtraction formulas?
2. Can you explain the properties of the cotangent function?
3. How would this proof change for other angle combinations, like $30^\circ$ or $45^\circ$?
4. What is the geometric interpretation of tangent and cotangent?
5. Can you demonstrate this identity using a unit circle?

Tip: Trigonometric identities often simplify to well-known numbers, so looking for patterns can save time in proofs.