Answer
$$\tan\theta+\cot\theta=\sec\theta\csc\theta$$
The proof is below.
Work Step by Step
$$\tan\theta+\cot\theta=\sec\theta\csc\theta$$
We take a look at the left side first.
$$X=\tan\theta+\cot\theta$$
- Quotient identities: $$\tan\theta=\frac{\sin\theta}{\cos\theta}\hspace{2cm}\cot\theta=\frac{\cos\theta}{\sin\theta}$$
Then, $$X=\frac{\sin\theta}{\cos\theta}+\frac{\cos\theta}{\sin\theta}$$
$$X=\frac{\sin^2\theta+\cos^2\theta}{\sin\theta\cos\theta}$$
- Pythagorean identities: $\sin^2\theta+\cos^2\theta=1$
$$X=\frac{1}{\sin\theta\cos\theta}$$
$$X=\frac{1}{\sin\theta}\times\frac{1}{\cos\theta}$$
- Finally, Reciprocal identities: $$\csc\theta=\frac{1}{\sin\theta}\hspace{2cm}\sec\theta=\frac{1}{\cos\theta}$$
Thus,
$$X=\csc\theta\sec\theta$$
So 2 sides are equal. The equation is an identity as a result.
