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UPSC NDA 2 2022 Mathematics Solved Paper

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Question : 51 of 120

Marks: +1, -0

What is cos 36° − cos 72° equal to ?

$\frac{\sqrt{5}}{2}$
$-\frac{\sqrt{5}}{2}$
$\frac{1}{2}$
$-\frac{1}{2}$

Solution:

Concept:

Use the identity

\cos C - \cos D = -2 \sin\frac{C+D}{2} \sin\frac{C-D}{2}

and known exact values of

\sin 54^\circ

and

\sin 18^\circ

.

Explanation:

Let

C = 36^\circ

and

D = 72^\circ

.

Apply the identity:

\cos 36^\circ - \cos 72^\circ = -2 \sin\frac{36^\circ+72^\circ}{2} \sin\frac{36^\circ-72^\circ}{2}

.

This gives

\cos 36^\circ - \cos 72^\circ = -2 \sin 54^\circ \sin(-18^\circ)

.

Since

\sin(-\theta) = -\sin\theta

, the expression simplifies to

2 \sin 54^\circ \sin 18^\circ

.

Use known values:

\sin 54^\circ = \frac{\sqrt5+1}{4}

and

\sin 18^\circ = \frac{\sqrt5-1}{4}

.

Multiply:

2 \times \frac{\sqrt5+1}{4} \times \frac{\sqrt5-1}{4} = 2 \times \frac{(\sqrt5)^2 - 1^2}{16} = 2 \times \frac{5-1}{16} = 2 \times \frac{4}{16} = \frac{8}{16} = \frac12

.

Thus

\cos 36^\circ - \cos 72^\circ = \frac12

.

Answer:

\frac12

(Option C).

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