### Question 8

SSC-CGL 2020 Mar 3 Shift 2

If 2 sinθ + 15 cos²θ = 7, and θ is an acute angle, then the value of (tanθ + cosecθ + secθ) is?

### Solution in Brief

Write the given equation as 2 sinθ + 15 (1 - sin²θ) = 7, solve the quadratic equation to obtain sinθ = 4/5, and by pythagoras tanθ = 4/3, cosecθ = 5/4, secθ = 5/3, giving 17/4 ans!

### Solution in Detail

Given $\displaystyle 2\sin\theta + 15\cos^2\theta = 7$

Put $\displaystyle \cos^2 \theta = 1 - \sin^2 \theta$

$\displaystyle \therefore 2\sin\theta + 15(1 - \sin^2\theta) = 7$

$\displaystyle \therefore 15 \sin^2 \theta - 2\sin \theta - 8 = 0$

Solving the quadratic $\displaystyle \sin \theta = \frac 45$

By Pyth, $\displaystyle H = 5, P = 4, B = 3$

$\displaystyle \therefore \tan \theta = \frac{P}{B} = \frac43$

$\displaystyle \therefore \sec \theta = \frac{H}{B} = \frac53$

$\displaystyle \therefore \cosec \theta = \frac{H}{P} = \frac54$

$\displaystyle \therefore \tan \theta + \cosec \theta + \sec \theta$

$\displaystyle = \frac 43 + \frac 53 + \frac 54 = \frac{17}{4}\:\underline{Ans}$

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