Trigonometry problem

Problem:



Solution;

$\frac{\sec \theta}{\csc \theta - \cot \theta} - \frac{\sec \theta}{\csc \theta + \cot \theta}$

Multiply $\sin \theta \cos \theta$ above and below:

$\frac{\sin \theta}{\cos \theta - \cos^2 \theta} - \frac{\sin \theta}{\cos \theta + \cos^2 \theta}$

Make common denominator gives:

$\frac{\sin \theta(\cos \theta + \cos^2 \theta)}{(\cos \theta + \cos^2 \theta)(\cos \theta - \cos^2 \theta)} - \frac{\sin \theta(\cos \theta - \cos^2 \theta)}{(\cos \theta + \cos^2 \theta)(\cos \theta - \cos^2 \theta)}$

Now do the subtraction

$\frac{\sin \theta(\cos \theta + \cos^2 \theta)-\sin \theta(\cos \theta - \cos^2 \theta)}{(\cos \theta + \cos^2 \theta)(\cos \theta - \cos^2 \theta)}$

Simplify the numerator by just expand and subtract:

$\frac{2\sin \theta\cos^2 \theta}{(\cos \theta + \cos^2 \theta)(\cos \theta - \cos^2 \theta)}$

Simplify the denominator:

$\frac{2\sin \theta\cos^2 \theta}{\cos^2 \theta(1 + \cos \theta)(1 - \cos \theta)}$

Cancel the $\cos^2 \theta$:

$\frac{2\sin \theta}{ (1 + \cos \theta)(1 - \cos \theta)}$

Expand the denominator:

$\frac{2\sin \theta}{1  - \cos^2 \theta}$

Just an identity

$\frac{2\sin \theta}{\sin^2 \theta}$

There we go

$2 \csc \theta$