Question

If (cosecθ - sinθ) (secθ - cosθ) (tanθ + cotθ) - tanθ = 0, 0° < θ < 90°, then the value of \(\frac{2cosθ + sinθ}{5cosθ - sin θ}\) 
1. 3/4
2. 2/3
3. 5/5
4. 1

Answer 1

Correct Answer - Option 1 : 3/4

Given :

(cosecθ - sinθ) (secθ - cosθ) (tanθ + cotθ) - tanθ = 0

Formula used :

cosecθ = 1/sinθ, secθ = 1/cosθ, tanθ = sinθ/cosθ and cotθ = cosθ/sinθ 

Calculations :

(cosecθ - sinθ) (secθ - cosθ) (tanθ + cotθ) = tanθ 

[(1/sinθ) - sinθ) [(1/cosθ) - cosθ) [(sinθ/cosθ) + (cosθ/sinθ) = tanθ 

[(1 - sin²θ)/sinθ] [(1 - cos²θ)/cosθ] [(sin²θ + cos²θ)/sinθ.cosθ] = tanθ 

sin²θ + cos²θ = tanθ 

tanθ = 1 

θ = 45° 

Now 

(2cosθ + sinθ)/(5cosθ -sinθ) = [2(1/√2) + (1/√2)]/[5(1/√2) - (1/√2)]  

⇒ (3/√2)/(4/√2)

⇒ 3/4 

∴ The required value of the required equation is 3/4