Question

What is (1 + cot θ - cosec θ)(1 + tan θ + sec θ) equal to?

1. 1
2. 2
3. 3
4. 4

Answer 1

Correct Answer - Option 2 : 2

Concept:

Trigonometry Formula

cot θ = \(\rm \frac{cos θ }{sin θ }\)

tan θ = \(\rm \frac{sin θ }{cos θ }\)

cosec θ = \(\rm \frac{1 }{sin θ }\)

sec θ = \(\rm \frac{1}{cos θ }\)

sin² θ + cos² θ = 1

(sin θ + cos θ)² = sin² θ + cos² θ + 2 sin θ cos θ 

 

Calculation:

To Find: Value of (1 + cot θ - cosec θ)(1 + tan θ + sec θ)

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

= \(\rm \frac{(sin θ + cos θ - 1) ( sin θ + cos θ + 1)}{sin θ \: cosθ }\)

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

= \(\rm \frac{sin^{2}θ + cos^{2}θ + 2 sin θ cos θ - 1}{sinθ \: cos θ }\)

= \(\rm \frac{1 + 2 sin θ cos θ - 1}{sinθ \: cos θ }\)

= \(\rm \frac{2 sin θ cos θ }{sinθ \: cos θ }\)

= 2