Q.Integrate the following function: $\int \frac{\sec^2 x}{\operatorname{cosec}^2 x} dx$

The key idea is to rewrite the integrand in terms of sine and cosine, then use the sine double-angle identity.

First, rewrite using $\sec x = \frac{1}{\cos x}$ and $\operatorname{cosec} x = \frac{1}{\sin x}$:

$\frac{\sec^2 x}{\operatorname{cosec}^2 x} = \frac{1/\cos^2 x}{1/\sin^2 x} = \frac{\sin^2 x}{\cos^2 x} = \tan^2 x.$ …

The key here is to first rewrite the integrand in terms of sines and cosines. The expression $\frac{\sec^2 x}{\operatorname{cosec}^2 x}$ looks complicated, but it’s just a ratio of two reciprocal functions. Let’s simplify it before integrating.

Recall:

• $\sec x = \frac{1}{\cos x}$, so $\sec^2 x = \frac{1}{\cos^2 x}$.
• $\operatorname{cosec} x = \frac{1}{\sin x}$, so $\operatorname{cosec}^2 x = \frac{1}{\sin^2 x}$.

Thus:

So the integral becomes:

Now, to integrate $\tan^2 x$, we use the Pythagorean identity:

This is a standard trick: whenever you see $\tan^2 x$ or $\cot^2 x$, rewrite them using $\sec^2$ or $\operatorname{cosec}^2$ to get a simple integral.

So:

Now integrate term by term:

1. $\int \sec^2 x \, dx = \tan x + C_1$ (since the derivative of $\tan x$ is $\sec^2 x$). …

Method: Simplify to a single trig function, then use a Pythagorean identity

Use this when a ratio of reciprocal trig functions ($\sec, \csc, \tan, \cot$) hides a much simpler expression — first simplify, then integrate.

Steps

Step 1: Convert everything to sine and cosine and cancel.

Writing $\sec x=1/\cos x$ and $\csc x=1/\sin x$ turns a ratio like $\dfrac{\sec^2 x}{\csc^2 x}$ into $\dfrac{\sin^2 x}{\cos^2 x}=\tan^2 x$.

Step 2: Replace a squared function using a Pythagorean identity.

There is no direct formula for $\int\tan^2 x\,dx$, but the identity …

Common Mistakes

Mistake 1: Inverting the ratio while simplifying.

Why it's wrong: $\dfrac{\sec^2 x}{\csc^2 x}=\dfrac{1/\cos^2 x}{1/\sin^2 x}=\dfrac{\sin^2 x}{\cos^2 x}=\tan^2 x$, not $\cot^2 x$ — dividing by a reciprocal flips it. Correct approach: multiply by the reciprocal of the bottom carefully.

Mistake 2: Trying to integrate $\tan^2 x$ directly. …