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UPSC NDA I 2021 Mathematics Solved Paper

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Question : 32 of 120

Marks: +1, -0

What is the value of the following?

\csc\left(\frac{7\pi}{6}\right)\sec\left(\frac{5\pi}{3}\right)

$\frac{4}{3}$
4
$-4$
$-\frac{4}{\sqrt{3}}$

Solution:

Concept:

Use trigonometric reduction formulas to evaluate angles outside the first quadrant.

Explanation:

Convert radians to degrees:

\csc\left(\frac{7\pi}{6}\right) = \csc(210^\circ)

\sec\left(\frac{5\pi}{3}\right) = \sec(300^\circ)

Apply identities:

\csc(180^\circ + \theta) = -\csc \theta

and

\sec(360^\circ - \theta) = \sec \theta

So

\csc(210^\circ) = \csc(180^\circ + 30^\circ) = -\csc 30^\circ

\csc 30^\circ = 2

, therefore

-\csc 30^\circ = -2

\sec(300^\circ) = \sec(360^\circ - 60^\circ) = \sec 60^\circ

\sec 60^\circ = 2

Multiply:

(-2) \times 2 = -4

Answer:

-4

(Option C).

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