Question

Steps on how to establish the identity:
(cos²θ/cosθ-sinθ) + (sin²θ/sinθ-cosθ) = sinθ+cosθ

How can I simplify the left side even further?

Answer 1

\[\large \frac{\cos^2\theta}{\cos \theta-\sin \theta}+\frac{\sin^2\theta}{\sin \theta-\cos \theta}=\sin \theta+\cos \theta\]

Let's start by factoring out a negative -1 from the right fraction.
This might not be immediately obvious why I did this.
Let me explain:

We want to be able to add these fractions together.
To do so, we need to have a common denominator.\[\large \frac{\cos^2\theta}{\cos \theta-\sin \theta}+\frac{\sin^2\theta}{\sin \theta-\cos \theta} \qquad \rightarrow \qquad \frac{\cos^2\theta}{\cos \theta-\sin \theta}+\frac{\sin^2\theta}{-(\cos \theta-\sin \theta)}\]

\[\large \rightarrow \qquad \frac{\cos^2\theta}{\cos \theta-\sin \theta}-\frac{\sin^2\theta}{\cos \theta-\sin \theta}\]

Understand what I did there?
That step can be a little tricky.

Answer 2

@zepdrix Thank you so much, I got it! My textbook completely skipped this step in the answer key, so I didn't really know how to proceed at that point. Thank you so much! :)

Answer 3

Oh cool! c: