Help to prove each Identity

Hello, AdamEdward!

Quote:

$(a)\;\;\tan\theta - 1 \:=\: \frac{\sin^2\!\theta - \cos^2\!\theta}{\sin\theta\cos\theta + \cos^2\!\theta}$

Factor the right side: . $\frac{(\sin\theta - \cos\theta)(\sin\theta + \cos\theta)}{\cos\theta(\sin\theta + \cos\theta)}$

Reduce: . $\frac{\sin\theta - \cos\theta}{\cos\theta} \;\;=\;\;\frac{\sin\theta}{\cos\theta} - \frac{\cos\theta}{\cos\theta} \;\;=\;\; \tan\theta - 1$

Quote:

$(b)\;\;\sin x + \sin x\cot^2\!x \:=\: \csc x$

Factor the left side: . $\sin x\cdot\underbrace{(1 + \cot^2\!x)}_{\text{This is }\csc^2\!x} \;\;=\;\;\sin x\cdot\csc^2\!x \;\;=\;\;\frac{1}{\csc x}\!\cdot \csc^2\!x \;\;=\;\;\csc x$