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UPSC NDA I 2019 Mathematics Solved Paper

Question : 32 of 120

Marks: +1, -0

Tan 54° can be expressed as

Solution:

Concept:

Use the tangent addition formula:

\tan(A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}

Explanation:

Write

54^\circ = 45^\circ + 9^\circ

Then

\tan 54^\circ = \tan(45^\circ+9^\circ) = \frac{\tan45^\circ + \tan9^\circ}{1-\tan45^\circ\tan9^\circ}

Since

\tan45^\circ = 1

, this simplifies to

\frac{1+\tan9^\circ}{1-\tan9^\circ}

Multiply numerator and denominator by

\cos9^\circ

\frac{(1+\tan9^\circ)\cos9^\circ}{(1-\tan9^\circ)\cos9^\circ} = \frac{\cos9^\circ + \sin9^\circ}{\cos9^\circ - \sin9^\circ}

Therefore,

\tan54^\circ

is expressed as

\frac{\cos9^\circ + \sin9^\circ}{\cos9^\circ - \sin9^\circ}

Answer:

\frac{\cos9^\circ + \sin9^\circ}{\cos9^\circ - \sin9^\circ}

(Option C)

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