The number of solution of tanx+secx=2cosx\; in\; \left [ 0,2\pi \right )\; is

  • Option 1)

    2

  • Option 2)

    3

  • Option 3)

    0

  • Option 4)

    1

 

Answers (1)

As we learnt in 

Trigonometric Equations -

The equations involving trigonometric function of unknown angles are known as trigonometric equations.

- wherein

e.g. \cos ^{2}\Theta - 4\cos \Theta = 1

 

 \tan x+\sec x=2\cos x --(i)

=\frac{\sin x+1}{\cos x}=2\cos x

multiplying both sides by \cos x, we get

\Rightarrow 1+\sin x=2\cos ^{2}x=2-2\sin ^{2}x

\Rightarrow 2\sin ^{2}x+\sin x-1=0

\sin x=\frac{-1\pm \sqrt{1+8}}{4}=\frac{-1\pm {3}}{4} =-1\ or\ \frac{1}{2}

now   \sin x=\frac{1}{2} given from solution x=\frac{\pi}{6},\frac{5\pi}{6}

\sin x=-1\Rightarrow cos x=0 \ and \ x=\frac{3\pi}{2}

now L.H.S of equ.(i) when   x=\frac{3\pi}{2}

=\lim_{x\rightarrow \frac{3\pi}{2}}(\tan x+\sec x)=\lim_{x\rightarrow \frac{3\pi}{2}}(\frac{1+\sin x}{\cos x})=\lim_{x\rightarrow \frac{3\pi}{2}} \tfrac{-cos x}{\sin x}=0\ R.H.S

Thus x=\frac{\pi}{6}, \frac{5\pi}{6}\ or\ \frac{3\pi}{6} \ are\ 3 \ solution


Option 1)

2

This is incorrect option

Option 2)

3

This is correct option

Option 3)

0

This is incorrect option

Option 4)

1

This is incorrect option

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