\(\int\frac{\cos2x}{\cos^2x.\sin^2x}dx= \ ?\)

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  1. -cot x - tan x + c
  2. cot x - tan x + c
  3. cot x + tan x + c
  4. tan x - cot x + c

Answer (Detailed Solution Below)

Option 1 : -cot x - tan x + c
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Detailed Solution

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Concept:

  • cos 2x = cosx - sinx

Calculation:

\(\int\frac{\cos2x}{\cos^2x.\sin^2x}dx\)

\(\int\frac{cos^{2}x-sin^{2}x}{\cos^2x.\sin^2x}dx \)

\(\int \left ( \frac{1}{sin^{2}x}-\frac{1}{cos^{2}x} \right )dx\)

\(\int \frac{1}{\sin^{2}x}dx-\int \frac{1}{\cos^{2}x} dx\)

\(\rm \int cosec^{2}xdx-\int sec^{2}x dx\)

= - cot x - tan x + C

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