If tan θ/2 = cosec θ - sin θ , then ² θ/2 =

Correct Answer is : - 2 + 5{5}5

Correct Answer is : - 2 + 5{5}5

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AP EAMCET Engineering 22 Apr 2019 Shift 1 Paper

Section: Mathematics

Question : 22 of 160

Marks: +1, -0

If tan

\frac{\theta}{2}

\tan^2 \frac{\theta}{2}

Solution:

Simplify the given expression,

\tan \frac{\theta}{2} = \csc \theta - \sin \theta

\frac{1}{\sin \theta} - \sin \theta

= \frac{1 - \sin^2 \theta}{\sin \theta}

= \frac{\cos^2 \theta}{\sin \theta}

= \cot \theta \cos \theta

= \frac{\left(1 - \tan^2 \frac{\theta}{2}\right)^{2}}{2 \tan \frac{\theta}{2} \left(1 + \tan^2 \frac{\theta}{2}\right)}

Further simplify the above,

2 \tan^2 \frac{\theta}{2} \left(1 + \tan^2 \frac{\theta}{2}\right) = 1 + \tan^4 \frac{\theta}{2}

-2 \tan^2 \frac{\theta}{2}

2 \tan^2 \frac{\theta}{2} + 2 \tan^4 \frac{\theta}{2} = 1 + \tan^4 \frac{\theta}{2}

-2 \tan^2 \frac{\theta}{2}

\tan^4 \frac{\theta}{2} + 4 \tan^2 \frac{\theta}{2} - 1 = 0

Solving the above quadratic equation, we get

\tan^2 \frac{\theta}{2} = \frac{-4 \pm \sqrt{4^2 - (-1) 4}}{2(1)}

= -2 \pm \sqrt{5}

= -2 + \sqrt{5}

\left[ \because \tan^2 \frac{\theta}{2} > 0 \right]

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