# How do you prove (1 + cot²θ) / secθ = cscθcotθ?

Jul 17, 2016

For this particular problem, we can make use of a trigonometric identity, namely

$1 + {\cot}^{2} \left(\theta\right) = {\csc}^{2} \left(\theta\right)$

Modifying the left-hand side, we can write

$\frac{{\csc}^{2} \left(\theta\right)}{\sec \left(\theta\right)} = \csc \left(\theta\right) \cot \left(\theta\right)$

Now, we can use the reciprocal identities of $\csc \left(\theta\right)$, $\sec \left(\theta\right)$ and $\cot \left(\theta\right)$, which are the following:

$\csc \left(\theta\right) = \frac{1}{\sin} \left(\theta\right)$

$\sec \left(\theta\right) = \frac{1}{\cos} \left(\theta\right)$

$\cot \left(\theta\right) = \cos \frac{\theta}{\sin} \left(\theta\right)$

Replacing the left-hand side with these reciprocal identities we get

$\frac{\frac{1}{\sin} ^ 2 \left(\theta\right)}{\frac{1}{\cos} \left(\theta\right)} = \csc \left(\theta\right) \cot \left(\theta\right)$

To make the left-hand side a lot easier, we can multiply both the numerator and denominator by the reciprocal of the denominator, namely $\cos \left(\theta\right)$, giving us

$\frac{\frac{1}{\sin} ^ 2 \left(\theta\right) \cdot \cos \frac{\theta}{1}}{\cancel{\frac{1}{\cos} \left(\theta\right) \cdot \cos \frac{\theta}{1}}} = \csc \left(\theta\right) \cot \left(\theta\right)$

$\frac{\cos \left(\theta\right)}{{\sin}^{2} \left(\theta\right)} = \csc \left(\theta\right) \cot \left(\theta\right)$

By converting the left-hand side back to it's original identities, we can conclude that it does, in fact, equal to the right-hand side.

$\frac{1}{\sin} \left(\theta\right) \cdot \frac{\cos \left(\theta\right)}{\sin \left(\theta\right)} = \csc \left(\theta\right) \cot \left(\theta\right)$

$\csc \left(\theta\right) \cot \left(\theta\right) = \csc \left(\theta\right) \cot \left(\theta\right)$

Thus, we have verified this equation.

#### Explanation:

I recommend modifying one side only, just so it is much clearer as to what is going on in the first place. These problems mostly require trigonometric identities and reciprocal identities, so make sure you know how to use them.