Answer: \(\tan\theta+\cot\theta=\sec\theta\operatorname{cosec}\theta\).

Use the standard trigonometric identities and make sure the final angles are chosen from the interval given in the question.

Write each term using sine and cosine:

\(\tan\theta+\cot\theta=\frac{\sin\theta}{\cos\theta}+\frac{\cos\theta}{\sin\theta}.\)

Using a common denominator,

\(\frac{\sin^2\theta+\cos^2\theta}{\sin\theta\cos\theta}=\frac{1}{\sin\theta\cos\theta}.\)

Since \(\sec\theta=\frac{1}{\cos\theta}\) and \(\operatorname{cosec}\theta=\frac{1}{\sin\theta}\),

\(\frac{1}{\sin\theta\cos\theta}=\sec\theta\operatorname{cosec}\theta.\)

This gives the required answer: \(\tan\theta+\cot\theta=\sec\theta\operatorname{cosec}\theta\).