Answer
$$\frac{\sin\theta+\tan\theta}{1+\cos\theta}=\tan\theta$$
The process of identity verification is shown in the work step by step.
Work Step by Step
$$\frac{\sin\theta+\tan\theta}{1+\cos\theta}=\tan\theta$$
We examine the left side.
$$X=\frac{\sin\theta+\tan\theta}{1+\cos\theta}$$
- First, we rewrite $\tan\theta$ into $\frac{\sin\theta}{\cos\theta}$, as in Quotient Identity.
$$X=\frac{\sin\theta+\frac{\sin\theta}{\cos\theta}}{1+\cos\theta}$$
$$X=\frac{\frac{\sin\theta\cos\theta+\sin\theta}{\cos\theta}}{1+\cos\theta}$$
$$X=\frac{\sin\theta\cos\theta+\sin\theta}{\cos\theta(1+\cos\theta)}$$
$$X=\frac{\sin\theta(\cos\theta+1)}{\cos\theta(\cos\theta+1)}$$
$$X=\frac{\sin\theta}{\cos\theta}$$
- Now, in reverse, $\frac{\sin\theta}{\cos\theta}$ can be written into $\tan\theta$.
$$X=\tan\theta$$
That means, $$\frac{\sin\theta+\tan\theta}{1+\cos\theta}=\tan\theta$$
The equation is an identity as a result.
