Hello, Mike!

You seem to have the basics that are needed.

You should learn to "clear" the complex fractions . . .

Quote:

$\displaystyle (e)\;\;\frac{1 + \sin\theta}{1 + \csc\theta} \:=\:\sin\theta$

The left side is: .$\displaystyle \frac{1 + \sin\theta}{1 + \frac{1}{\sin\theta}}$

Multiply by $\displaystyle \frac{\sin\theta}{\sin\theta}\!:\;\;\frac{\sin\the ta(1 + \sin\theta)}{\sin\theta\left(1 + \frac{1}{\sin\theta}\right)} \;=\;\frac{\sin(1 + \sin\theta)}{\sin\theta + 1} \;=\;\sin\theta$

Quote:

$\displaystyle (f)\;\;\frac{\sin\theta + \tan\theta}{\cos\theta + 1} \;=\;\tan\theta$

The left side is: .$\displaystyle \frac{\sin\theta + \frac{\sin\theta}{\cos\theta}}{\cos\theta + 1} \;=\;\frac{\sin\theta\left(1 + \frac{1}{\cos\theta}\right)}{\cos\theta + 1} $

Multiply by $\displaystyle \frac{\cos\theta}{\cos\theta}\!:\;\;\frac{\cos\the ta\cdot\sin\theta\left(1 + \frac{1}{\cos\theta}\right)}{\cos\theta(\cos\theta + 1)} \;=\;\frac{\sin\theta(\cos\theta + 1)}{\cos\theta(\cos\theta + 1)}$

. . . . . . $\displaystyle = \;\frac{\sin\theta}{\cos\theta} \;=\;\tan\theta $

Quote:

$\displaystyle 4(b)\;\;\csc^2\!\theta - 1 \;=\;\csc^2\!\theta\cos^2\!\theta$

The left side is a basic identity .$\displaystyle \csc^2\!\theta-1 \;=\;\cot^2\!\theta$

Then: .$\displaystyle \cot^2\!\theta \;=\;\frac{\cos^2\!\theta}{\sin^2\!\theta} \;=\;\frac{1}{\sin^2\!\theta}\cdot\cos^2\!\theta \;=\;\csc^2\!\theta\cos^2\!\theta$

Quote:

$\displaystyle (c)\;\;\sin^2\!\theta \;=\;\frac{\tan^2\!\theta}{1 + \tan^2\!\theta} $

The right side is: .$\displaystyle \frac{\frac{\sin^2\!\theta}{\cos^2\!\theta}}{1 + \frac{\sin^2\!\theta}{\cos^2\!\theta}} $

Multiply by $\displaystyle \frac{\cos^2\!\theta}{\cos^2\!\theta}\!:\;\;\frac{ \cos^2\!\theta\left(\frac{\sin^2\!\theta}{\cos^2\! \theta}\right)}{\cos^2\!\theta\left(1 + \frac{\sin^2\!\theta}{\cos^2\!\theta}\right)}$

. . . . . $\displaystyle = \;\;\frac{\sin^2\!\theta}{\underbrace{\cos^2\!\the ta + \sin^2\!\theta}_{\text{This is 1}}} \;\;=\;\;\sin^2\!\theta $

Edit: While I was typing up this long disseration, Jonboy did a great job!

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