What is {{ θ }}{{1 + θ }} + {1}{{ θ }} equal to?

Correct Answer is : secθsec θsecθ

Correct Answer is : secθsec θsecθ

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UPSC CDS 1 2020 Math Paper

Question : 59 of 100

Marks: +1, -0

What is

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

equal to?

Solution:

Concept:

Simplify the given trigonometric expression by converting all terms to sine and cosine, then combine and reduce using

\sin^2\theta + \cos^2\theta = 1

Explanation:

Step 1: Rewrite

\frac{1}{\cot\theta}

\tan\theta = \frac{\sin\theta}{\cos\theta}

So the expression becomes

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

Step 2: Take the common denominator

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

Numerator:

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

Step 3: Use

\sin^2\theta + \cos^2\theta = 1

to get

1 + \sin\theta

Thus the expression equals

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

Step 4:

\frac{1}{\cos\theta} = \sec\theta

Therefore, the given expression simplifies to

\sec\theta

Answer:

\sec\theta

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