Question

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation cos^–1 (x) – 2sin^–1(x) = cos^–1 (2x) is equal to ______.

पर्याय

• 0

• 1

• `1/2`

• `-1/2`

Answer 1

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation cos^–1 (x) – 2sin^–1(x) = cos^–1 (2x) is equal to 0.

Explanation:

We are given that

cos^–1 x – 2 sin^–1 x = cos^–1 2x

`\implies cos^-1x - 2(π/2 - cos^-1x) = cos^-1 2x`

`\implies` cos^–1 x – π + 2cos^–1 x = cos^–1 2x

`\implies` 3cos^–1 x = π + cos^–1 2x  ...(i)

`\implies` cos(3cos^–1 x) = cos(π + cos^–1 2x)  ...[∵ 3cos^–1 x = cos^–1 (4x³ – 3x)]

`\implies` 4x³ – 3x = –2x

`\implies` 4x³ = x

`\implies` x = `0, ±1/2`

Here all values of x satisfy the equation (A)

∴ Sum of all the solutions of the equation = `-1/2 + 1/2 + 0` = 0