If $ \theta $ is a positive acute angle such that $ \sec \theta=\operatorname{cosec} 60^{\circ} $, find the value of $ 2 \cos ^{2} \theta-1 $.

Given:

\( \theta \) is a positive acute angle such that \( \sec \theta=\operatorname{cosec} 60^{\circ} \).

To do:

We have to find the value of \( 2 \cos ^{2} \theta-1 \).

Solution:  

We know that,

$\operatorname{cosec}\ (90^{\circ}- \theta) = sec\ \theta$

Therefore,

$\sec \theta=\operatorname{cosec}\ (90^{\circ}- \theta)$

$\Rightarrow \operatorname{cosec}\ (90^{\circ}- \theta)=\operatorname{cosec}\ 60^{\circ}$

Comparing on both sides, we get,

$90^{\circ}- \theta=60^{\circ}$

$\theta=90^{\circ}-60^{\circ}$

$\theta=30^{\circ}$

This implies,

$2 \cos ^{2} \theta-1=2 \cos ^{2}30^{\circ}-1$

$=2(\frac{\sqrt3}{2})^2-1$

$=2(\frac{3}{4})-1$

$=\frac{3}{2}-1$

$=\frac{1}{2}$

The value of \( 2 \cos ^{2} \theta-1 \) is $\frac{1}{2}$. 

Tutorialspoint

Updated on: 10-Oct-2022

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