If sec θ = x + 1 4 x , x ≠ 0 , find sec θ + tan θ .

Given sec θ = x + 1 4 x . We know that tan θ = sec 2 θ - 1 = x + 1 4 x 2 - 1 = x 2 + 2 × x × 1 4 x + 1 4 x 2 - 1 = x 2 + 1 2 + 1 16 x 2 - 1 = x 2 - 1 2 + 1 16 x 2 = x 2 - 2 × x × 1 4 x + 1 16 x 2 = x - 1 4 x 2 = ± x - 1 4 x If tan θ = x - 1 4 x , sec θ + tan θ = x + 1 4 x + x - 1 4 x = 2 x . If tan θ = - x - 1 4 x , sec θ + tan θ = x + 1 4 x - x - 1 4 x = x + 1 4 x - x + 1 4 x = 1 2 x . Therefore, the required value of sec θ + tan θ is 2 x or 1 2 x .

If sec θ - tan θ = x , show that sec θ + tan θ = 1 x and hence find the values of cos θ and sin θ .

Given that, sec θ - tan θ = x -----(i) and sec θ + tan θ = 1 x -----(ii) Adding (i) with (ii), we obtain, sec θ - tan θ + sec θ + tan θ = x + 1 x ⇒ 2 sec θ = x + 1 x ⇒ 2 cos θ = x 2 + 1 x ⇒ cos θ x 2 + 1 = 2 x ⇒ cos θ = 2 x x 2 + 1 . Now, we have, sin θ = 1 - cos 2 θ = 1 - 2 x x 2 + 1 2 = 1 - 4 x 2 x 4 + 2 x 2 + 1 = x 4 + 2 x 2 + 1 - 4 x 2 x 4 + 2 x 2 + 1 = x 4 - 2 x 2 + 1 x 4 + 2 x 2 + 1 = x 2 - 1 2 x 2 + 1 2 = x 2 - 1 x 2 + 1 Therefore, the required value of sin θ is x 2 - 1 x 2 + 1 and cos θ is 2 x x 2 + 1 .

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CBSE Syllabus Standard Mathematics for Class X>Chapter 11 - Trigonometric Identities>Exercise 11.1

M L Aggarwal Solutions for Chapter: Trigonometric Identities, Exercise 1: Exercise 11.1

Author:M L Aggarwal

M L Aggarwal Mathematics Solutions for Exercise - M L Aggarwal Solutions for Chapter: Trigonometric Identities, Exercise 1: Exercise 11.1

Attempt the free practice questions on Chapter 11: Trigonometric Identities, Exercise 1: Exercise 11.1 with hints and solutions to strengthen your understanding. CBSE Syllabus Standard Mathematics for Class X solutions are prepared by Experienced Embibe Experts.