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  • What is the range of  <img alt="\frac{1+ \tan^2 \theta}{1-\tan^2 \theta}" src="https://learn.careers360.com/latex-image/?%5Cfrac%7B1&plus;%20%5Ctan%5E2%20%5Ctheta%7D%7B1-%5Ctan%5E2%20

What is the range of  \frac{1+ \tan^2 \theta}{1-\tan^2 \theta}

Option: 1

(-\infty,0]\cup [1, \infty)


Option: 2

(-\infty,\infty)


Option: 3

(-\infty,0]\cup[2, \infty)


Option: 4

(-\infty,-1]\cup[1, \infty)


Answers (1)

best_answer

Trigonometric Ratio of Submultiple of an Angle -

Trigonometric Ratio of Submultiple of an Angle

 

Trigonometric Ratio of θ in terms of θ/2 
\begin{aligned} \sin ( \theta) &=2 \sin \frac{\theta}{2} \cos \frac{\theta}{2} \\&=\frac{2\tan\frac{\theta}{2}}{1+\tan^2\frac{\theta}{2}} \\\cos ( \theta) &=\cos ^{2} \frac{\theta}{2}-\sin ^{2} \frac{\theta}{2} \\ &=1-2 \sin ^{2} \frac{\theta}{2} \\ &=2 \cos ^{2} \frac{\theta}{2}-1\\&=\frac{1-\tan^2\frac{\theta}{2}}{1+\tan^2\frac{\theta}{2}} \\ \tan ( \theta) &=\frac{2 \tan \frac{\theta}{2}}{1-\tan ^{2} \frac{\theta}{2}} \end{aligned}

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\frac{1+\tan^2 \theta}{1- \tan^2 \theta}=\frac {1+ \frac {\sin^2 \theta }{ \cos^2 \theta}}{1-\frac {\sin^2 \theta}{\cos^2 \theta}}\\ = \frac{ \cos^2 \theta + \sin^2 \theta }{ \cos^2 \theta - \sin^2 \theta }\\ =\frac{1}{\cos 2\theta} =sec 2\theta

So the range of this function is (-\infty,-1]\cup[1, \infty)

Posted by

Ritika Kankaria

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