Test Index

UPSC CDS 1 2020 Math Paper

© examsnet.com

Question : 60 of 100

Marks: +1, -0

What is

\frac{{\sin \theta - \cos \theta + 1}}{{\sin \theta + \cos \theta - 1}} - \frac{{\sin \theta + 1}}{{\cos \theta }}

equal to?

0
1
$2 sin θ$
$2 cos θ$

Solution:

Concept:

Rationalize the first fraction and then simplify to compare with the second.

Explanation:

Let the first fraction be

X = \frac{\sin\theta - \cos\theta + 1}{\sin\theta + \cos\theta - 1}

.

Multiply numerator and denominator by

(\sin\theta + \cos\theta + 1)

:

X = \frac{[(\sin\theta+1) - \cos\theta] \cdot [(\sin\theta+1) + \cos\theta]}{(\sin\theta + \cos\theta - 1)(\sin\theta + \cos\theta + 1)}

This becomes

X = \frac{(\sin\theta+1)^2 - \cos^2\theta}{(\sin\theta+\cos\theta)^2 - 1}

.

Expand numerator:

(\sin^2\theta + 2\sin\theta + 1) - (1 - \sin^2\theta) = 2\sin^2\theta + 2\sin\theta

.

Expand denominator:

(\sin^2\theta + \cos^2\theta + 2\sin\theta\cos\theta) - 1 = 1 + 2\sin\theta\cos\theta - 1 = 2\sin\theta\cos\theta

.

So

X = \frac{2\sin\theta(\sin\theta+1)}{2\sin\theta\cos\theta} = \frac{1+\sin\theta}{\cos\theta}

, provided

\sin\theta \neq 0

.

The second fraction is

Y = \frac{\sin\theta+1}{\cos\theta}

.

Therefore,

X - Y = \frac{1+\sin\theta}{\cos\theta} - \frac{\sin\theta+1}{\cos\theta} = 0

.

Answer:

A.

0

© examsnet.com

Go to Question:

Prev Question

Next Question