Answer
$$\frac{\tan^2t+1}{\tan t\csc^2t}=\tan t$$
The equation is an identity. The proof is explained in detail in the work step by step section below.
Work Step by Step
$$\frac{\tan^2t+1}{\tan t\csc^2t}=\tan t$$
The left side would be examined first.
$$X=\frac{\tan^2t+1}{\tan t\csc^2t}$$
Here we would consider the numerator and denominator separately.
*Numerator:
According to Pythagorean Identity: $$\tan^2t+1=\sec^2t$$
However, as $\sec t=\frac{1}{\cos t}$, we have
$$\tan^2t+1=\sec^2t=\frac{1}{\cos^2t}$$
*Denominator:
Apply the following identities: $$\tan t=\frac{\sin t}{\cos t}\hspace{2cm}\csc t=\frac{1}{\sin t}$$
we have $$\tan t\csc^2t=\frac{\sin t}{\cos t}\times\frac{1}{\sin^2t}$$
$$\tan t\csc^2t=\frac{1}{\sin t\cos t}$$
Combine the numerator and denominator back to $X$:
$$X=\frac{\frac{1}{\cos^2t}}{\frac{1}{\sin t\cos t}}$$
$$X=\frac{\sin t\cos t}{\cos^2t}$$
$$X=\frac{\sin t}{\cos t}$$
- As we know, from Quotient Identity, $\tan t=\frac{\sin t}{\cos t}$, so
$$X=\tan t$$
In conclusion, $$\frac{\tan^2t+1}{\tan t\csc^2t}=\tan t$$
The equation is an identity.
