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$$ I = \int\frac{1}{\sin^2(x)\cos^2(x)}$$ I have tried the following:
$$\sin^2x = \sin x \cdot \sin x = \frac{1}{2}(1 - \cos2x)$$ $$\cos^2x = \cos x \cdot \cos x = \frac{1}{2}(1 + \cos2x)$$
The integral becomes:
$$I = 4\int\frac{1}{1 - \cos^2(2x)} = 4\int\frac{1}{\sin^2(2x)}$$
Substitute $u = 2x \implies du = 2dx$
$$I = 2\int\frac{1}{\sin^2u} = -2\cot(u)$$
Plugging back x I get:
$$I = -2\cot(2x)$$
I tried plugging in the integral into an integral-calculator and the answer was: $\tan(x) - \cot(x)$. Can you help me identify what I did wrong?

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  • $\begingroup$ I tried plugging upper and lower bound to the integral and it turns out the answer is the same! $-2cot(2x)$ = $tanx- cotx$ $\endgroup$ – Radu Gabriel Aug 30 '19 at 11:57
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    $\begingroup$ Please use MathJax for trigonometric functions like \cos $\endgroup$ – sera Aug 30 '19 at 12:02
  • $\begingroup$ I will, I did not know about it, sorry! $\endgroup$ – Radu Gabriel Aug 30 '19 at 12:10
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Your result is correct. Here's another approach:

$$\sin2x=2\sin x\cos x\implies \sin^2x\cos^2x=\frac14\sin^22x$$

and since $\;(\cot x)'=-\csc^2x=-\frac1{\sin^2x}\;$ , we get

$$\int\frac{dx}{\sin^2x\cos^2x}=\frac42\int\frac{d(2x)}{\sin^22x}=-2\cot 2x+C$$

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$\tan(x) - \cot(x) = \tan(x) -\frac{1}{\tan(x)} = \frac{\tan^2(x) - 1}{\tan(x)}$

As $\tan(2x) = \frac{2\tan(x)}{1- \tan^2(x)}$

This implies $\tan(x) - \cot(x) = -2\cot(2x)$

Ur correct, actually both r same

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\begin{align*} I & = \int\frac{1}{\sin^2(x)\cos^2(x)} \, dx\\ &= \int\frac{\sin^2x+\cos^2x}{\sin^2(x)\cos^2(x)} \, dx\\ &= \int\frac{1}{\cos^2(x)} \, dx + \int\frac{1}{\sin^2(x)} \, dx\\ &= \int\sec^2(x) \, dx + \int\csc^2(x) \, dx\\ &=\tan x -\cot x+c \end{align*}

But this is same as your answer, so nothing wrong with what you arrived at.

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