Class 10 Math Sample Paper 2019 | Question 30

This 2019 CBSE class 10 Maths 4 mark question is from Trigonometry. Concept: Expressing trigonometric ratios in terms of sin θ and cos θ and solving a quadratic equation using the quadratic formula.

Question 30: If sec θ + tan θ = p, then find the value of cosec θ.

Video Explanation

Explanatory Answer | CBSE sample paper 2019 Question 30

Step 1: Rewrite sec θ + tan θ = p in terms of sin θ and cos θ

sec θ + tan θ = p
\\frac{1}{\text{cos θ}})+ \\frac{\text{sin θ}}{\text{cos θ}}) = p
\\frac{\text{1 + sin θ }}{\text{cos θ}}) = p
1 + sin θ = p(cos θ)

Substitute cos θ = \\sqrt{1 – {sin}^{2}θ})
1 + sin θ = p\\sqrt{1 – {sin}^{2}θ})
Square both sides of the equation: (1 + sin θ)² = p²(1 - sin²θ)
1 + sin² θ + 2 sin θ = p² - p² sin² θ
(1 + p²) sin² θ + 2 sin θ + (1 – p²) = 0

Step 2: Solve the quadratic equation for sin θ

Roots of a quadratic equation ax² + bx + c = 0 are,

\\frac{−b ± \sqrt{b^2 - 4ac}}{2a})

Compute Discriminant, D: D = b² - 4ac
In the equation, (1 + p²) sin² θ + 2 sin θ + (1 – p²) = θ, a = (1 + p²); b = (2); c = (1 – p²)
Discriminant, D = 4 – 4(1 + p²)(1 – p²)
= 4 – 4(1 – p⁴) = 4p⁴

Substitute the value of D,
The root, sin θ = \\frac{−2 ± √4p^4}{ 2(1 + p^2)}) = \\frac{−1 ± p^2}{(1 + p^2)}) = \\frac{p^2 − 1}{p^2 + 1}) , -1
cosec θ = \\frac{\text{1}}{\text{sin θ}}) = \\frac{p^2 + 1}{p^2 − 1}) , -1