Question

Let x = `(sin^3θ)/(cos^2θ)`, y = `(cos^3θ)/(sin^2θ)` and sinθ + cosθ = `1/2`. If x + y = `p/q` where p and q are coprime then (p + q) is equal to ______.

Options

• 96.00

• 97.00

• 98.00

• 99.00

Answer 1

Let x = `(sin^3θ)/(cos^2θ)`, y = `(cos^3θ)/(sin^2θ)` and sinθ + cosθ = `1/2`. If x + y = `p/q` where p and q are coprime then (p + q) is equal to 97.00.

Explanation:

x = `(sin^3θ)/(cos^2θ) = (sinθ(1 - cos^2θ))/(cos^2θ)`

x = `sinθ/(cos^2θ) - sinθ`, y = `cosθ/(sin^2θ) - cosθ`

x + y = `((sin^3θ + cos^3θ))/(cos^2θsin^2θ) - 1/2`

(x + y) = `(1/2(1 - sinθcosθ))/((sin2θ)^2/4) - 1/2`

= `(2(1 - (sin2θ)/2))/(sin^2 2θ) - 1/2`

Given that sinθ + cosθ = `1/2`

⇒ sin2θ = `-3/4`

⇒ `(2(1 + 3/8))/(9/16) - 1/2`

⇒ `44/9 - 1/2`

⇒ `79/18 = p/q`

⇒ p + q = 97