Prove that $$ an heta+\cot heta=\frac{1}{\cos heta \sin heta} $$

Question

Prove that $$ 	an 	heta+\cot 	heta=\frac{1}{\cos 	heta \sin 	heta} $$

UpStudy AI · Accepted Answer

We start with the left-hand side of the equation:

1. Write the definitions of $$ \tan \theta $$ and $$ \cot \theta $$:

$$
   \tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta}
   $$

2. So, the sum becomes:

$$
   \tan \theta + \cot \theta = \frac{\sin \theta}{\cos \theta} + \frac{\cos \theta}{\sin \theta}
   $$

3. To add these two fractions, find a common denominator which is $$ \cos \theta \sin \theta $$:

$$
   \tan \theta + \cot \theta = \frac{\sin^2 \theta}{\cos \theta \sin \theta} + \frac{\cos^2 \theta}{\cos \theta \sin \theta}
   $$

4. Combine the fractions:

$$
   \tan \theta + \cot \theta = \frac{\sin^2 \theta + \cos^2 \theta}{\cos \theta \sin \theta}
   $$

5. Use the Pythagorean identity, which states $$ \sin^2 \theta + \cos^2 \theta = 1 $$:

$$
   \tan \theta + \cot \theta = \frac{1}{\cos \theta \sin \theta}
   $$

Thus, we have proven that:

$$
\tan \theta + \cot \theta = \frac{1}{\cos \theta \sin \theta}
$$
$$ \tan \theta + \cot \theta = \frac{1}{\cos \theta \sin \theta} $$

Prove that \( \tan \theta+\cot \theta=\frac{1}{\cos \theta \sin \theta} \)