Prove that y = 4 sinθ/(2 + cosθ) - θ is an increasing function of θ in [0, π/2]

Solution:

We have,

y = 4 sin θ / (2 + cosθ) - θ

Therefore,

dy/dθ = (2 + cosθ)(4 cosθ) - 4sinθ (- sinθ)/(2 + cosθ)² - 1

= (8 cosθ + 4 cos² θ + 4sin² θ)/(2 + cosθ)² - 1

= (8 cosθ + 4)/(2 + cosθ)² - 1

Now, dy/dθ = 0

Hence,

(8 cosθ + 4)/(2 + cosθ)² = 1

⇒ 8 cosθ + 4 = 4 + cos² θ + 4 cosθ

⇒ cos² θ - 4 cosθ = 0

⇒ cosθ (cosθ - 4) = 0

⇒ cosθ = 0 or cosθ = 4

Since, cosθ ≠ 4

Therefore,

cos θ = 0

⇒ θ = π/2

Now,

dy/dθ = (8 cosθ + 4 - (4 + cos² θ + 4 cosθ))/(2 + cosθ)²

= (4 cosθ - cos² θ)/(2 + cosθ)²

= (cos(4 - cosθ))/(2 + cosθ)²

In interval [0, π/2],

we have cosθ > 0

Also,

⇒ 4 > cosθ

⇒ 4 - cosθ > 0

Hence,

cosθ (4 - cosθ) > 0 and also (2 + cosθ)2 > 0

Therefore,

cosθ (4 - cosθ)/(2 + cosθ)² > 0

Hence, dy/dθ > 0

So, y is strictly increasing in [0, π/2] and the given function is continuous at x = 0 and x = π/2.

Thus, y is increasing in the interval [0, π/2]

---

NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.2 Question 9

Prove that y = 4 sinθ/(2 + cosθ) - θ is an increasing function of θ in [0, π/2]

Summary:

Hence we have proved that y = 4 sinθ/(2 + cosθ) - θ is an increasing function of θ in [0, π/2]