Answer
$$\csc\theta\cos^2\theta+\sin\theta=\csc\theta$$
The equation is an identity as proved below.
Work Step by Step
$$\csc\theta\cos^2\theta+\sin\theta=\csc\theta$$
Take a look at the left side.
$$X=\csc\theta\cos^2\theta+\sin\theta$$
- Reciprocal identity: $$\csc\theta=\frac{1}{\sin\theta}$$
Therefore,
$$X=\frac{1}{\sin\theta}\times\cos^2\theta+\sin\theta$$
$$X=\frac{\cos^2\theta}{\sin\theta}+\sin\theta$$
$$X=\frac{\cos^2\theta+\sin^2\theta}{\sin\theta}$$
- Pythagorean identity: $\cos^2\theta+\sin^2\theta=1$
So, $$X=\frac{1}{\sin\theta}$$
Again, the first reciprocal identity would be applied here to rewrite $X$ into $\csc\theta$.
$$X=\csc\theta$$
In conclusion, $$\csc\theta\cos^2\theta+\sin\theta=\csc\theta$$
The equation has been verified to be an identity.
