The expression (tan θ + cot θ) (sec θ + tan θ) (1 – sin θ), 0° < θ < 90°, is equal to:

Shortcut Trick

Given:

(tan θ + cot θ) (sec θ + tan θ) (1 – sin θ), 0° < θ < 90°

Concept Used:

Here, The exact value of θ is not given so we can put any value of theta within 0° to 90° but take that value on which none of the two options will be the same.

So, We will take θ = 30°, We will not take 45° because at 45° (1) and (2) options will be the same 

Calculation:

At θ = 30° the given equation will be 

(tan 30° + cot 30°) (sec 30° + tan 30°) (1 – sin 30°)

⇒ \((\frac{1}{\sqrt3} + \frac{\sqrt3}{1}) (\frac{2}{\sqrt3} + \frac{1}{\sqrt3} ) ( 1 - \frac{1}{2} )\)

⇒ \((\frac{4}{\sqrt3}) (\frac{3}{\sqrt3} ) ( \frac{1}{2} )\)

⇒ 2

Now, On putting θ = 30° in options we get 

cosec 30° = 2

∴ The correct answer is cosecθ.

Alternate Method 

Given:

(tan θ + cot θ) (sec θ + tan θ) (1 – sin θ), 0° < θ < 90°

Formula Used:

sin²θ + cos²θ = 1

1 - sin²θ = cos²θ 

1/sinθ = cosecθ 

Calculation:

⇒ \((\frac{sinθ}{cosθ} + \frac{cosθ}{sinθ}) (\frac{1}{cosθ} + \frac{sinθ}{cosθ} ) ( 1 - sinθ)\)

⇒ \((\frac{sin^2θ + cos^2θ}{sinθ.cosθ} ) (\frac{1 + sinθ}{cosθ}) ( 1 - sinθ)\)

⇒ \((\frac{1}{sinθ.cosθ} ) (\frac{1 - sin^2θ}{cosθ})\)

⇒ \((\frac{1}{sinθ} ) (\frac{cos^2θ }{cos^2θ})\)

⇒ cosec θ

∴ The correct answer is cosec θ.