## Trigonometry (11th Edition) Clone

$$\csc\theta\cos^2\theta+\sin\theta=\csc\theta$$ The equation is an identity as proved below.
$$\csc\theta\cos^2\theta+\sin\theta=\csc\theta$$ 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.