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UPSC CDS 1 2022 Math Paper

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Question : 13 of 100

Marks: +1, -0

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

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

equal to?

$\frac{1}{x}$
$x$
$1+x$
$x-1$

Solution: 👈: Video Solution

Shortcut Trick

Given

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

Take

\theta = 45^{\circ}

We get

x = \frac{1 - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}}{1 + \frac{1}{\sqrt{2}}} = \frac{\sqrt{2}}{\sqrt{2}+1} = 2 - \sqrt{2} \; \; \cdots

(i)

To find

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

(let)

Put the same

\theta = 45^{\circ}

here, we get

y = \frac{\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} - 1}{\frac{1}{\sqrt{2}}} = 2 - \sqrt{2} \; \; \cdots \; \text{ (ii)} \;

Form (i) and (ii), we get

x = y

\therefore

The value of

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

is

x

.

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