To prove that $\csc\theta + \cot\theta = \cot\frac{\theta}{2}$, let us go step-by-step:

Step 1: Recall Trigonometric Definitions

$\csc\theta = \frac{1}{\sin\theta}, \quad \cot\theta = \frac{\cos\theta}{\sin\theta}.$

Step 2: Combine $\csc\theta + \cot\theta$

$\csc\theta + \cot\theta = \frac{1}{\sin\theta} + \frac{\cos\theta}{\sin\theta}.$ Since the denominators are the same: $\csc\theta + \cot\theta = \frac{1 + \cos\theta}{\sin\theta}.$

Step 3: Use the Half-Angle Identity

The half-angle identity for $\cot\frac{\theta}{2}$ is: $\cot\frac{\theta}{2} = \frac{1 + \cos\theta}{\sin\theta}.$

Step 4: Compare Both Sides

From the expression above, we see: $\csc\theta + \cot\theta = \cot\frac{\theta}{2}.$

Conclusion

Thus, the equation is proved: $\csc\theta + \cot\theta = \cot\frac{\theta}{2}.$

Would you like further details or explanations?

Follow-Up Questions

1. What are some other trigonometric proofs that use half-angle identities?
2. Can you rewrite this proof using graphical methods?
3. How is this identity used in solving trigonometric equations?
4. What happens when $\theta = 90^\circ$ or $\theta = 0^\circ$?
5. How does this identity relate to other identities involving cotangent and cosecant?

Tip

Memorize half-angle identities—they simplify many trigonometric problems and proofs.