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 $