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UPSC NDA I 2021 Mathematics Solved Paper

Question : 39 of 120

Marks: +1, -0

What is

(1+\cot\theta-\csc\theta)(1+\tan\theta+\sec\theta)

Solution: 👈: Video Solution

Concept:

Trigonometry Formula

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

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

\csc\theta = \frac{1}{\sin\theta}

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

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

(\sin\theta+\cos\theta)^{2} = \sin^{2}\theta + \cos^{2}\theta + 2\sin\theta\cos\theta

Calculation:

To Find: Value of

(1+\cot\theta-\csc\theta)(1+\tan\theta+\sec\theta)

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

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

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

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

= \frac{1+2\sin\theta\cos\theta-1}{\sin\theta\cos\theta}

= \frac{2\sin\theta\cos\theta}{\sin\theta\cos\theta}

= 2

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