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Trigonometric Ratios and Identities

Section: Mathematics

Question : 12 of 69

Marks: +1, -0

The value of

\csc 10^{\circ} - \sqrt{3} \sec 10^{\circ}

Solution:

Concept:

Rewrite

\csc 10^\circ

and

\sec 10^\circ

in terms of sine and cosine, then use the sine subtraction formula

\sin(A-B) = \sin A\cos B - \cos A\sin B

to simplify.

Explanation:

Start with

\csc 10^\circ - \sqrt{3} \sec 10^\circ

Express as

\frac{1}{\sin 10^\circ} - \frac{\sqrt{3}}{\cos 10^\circ}

Combine into one fraction:

\frac{\cos 10^\circ - \sqrt{3} \sin 10^\circ}{\sin 10^\circ \cos 10^\circ}

Multiply numerator and denominator by 4:

4 \times \frac{\frac{1}{2}\cos 10^\circ - \frac{\sqrt{3}}{2}\sin 10^\circ}{2 \sin 10^\circ \cos 10^\circ}

Note that

\frac{1}{2}\cos 10^\circ - \frac{\sqrt{3}}{2}\sin 10^\circ = \sin 30^\circ \cos 10^\circ - \cos 30^\circ \sin 10^\circ = \sin(30^\circ - 10^\circ) = \sin 20^\circ

Also,

2 \sin 10^\circ \cos 10^\circ = \sin 20^\circ

Therefore, the expression simplifies to

4 \times \frac{\sin 20^\circ}{\sin 20^\circ} = 4

Answer:

4

(Option D)

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