Question

The sum of solutions of the equation `cosx/(1 + sinx) = |tan2x|, x∈(-π/2, π/2) - {π/4, -π/4}` is ______.

Options

• `-(11π)/30`

• `-(7π)/30`

• `-π/15`

• `π/10`

Answer 1

The sum of solutions of the equation `cosx/(1 + sinx) = |tan2x|, x∈(-π/2, π/2) - {π/4, -π/4}` is `underlinebb(-(11π)/30)`.

Explanation:

Case-I: `0 ≤ x < π/4` and `(-π)/2 < x < (-π)/4`

⇒ `cosx/(1 + sinx) = (2sinxcosx)/(cos^2x - sin^2x)`

As tan2x > 0  and tan2x = `(sin2x)/(1 - 2sin^2x)`

⇒ cosx(–4sin²x – 2sinx + 1) = 0

⇒ cosx = 0 and 4sin²x + 2sinx – 1 = 0

∴ sinx = `(-2 ± 2sqrt(5))/8 = (-1 ± sqrt(5))/4`,

cosx ≠ 0 as x ≠ ± `π/2`

⇒ x = `π/10, (-3π)/10`

Case-II: `x∈(-π/2, π/2)` and `x∈{(-π)/4 < x < 0)`

⇒ `cosx/(1 + sinx) = (-2sinxcosx)/(cos^2x - sin^2x)`, as tan2x < 0

⇒ cosx(1 + 2sinx)` = 0

⇒ cosx = 0 and sinx = `(-1)/2`

⇒ x = `(-π)/6`

Sum of solutions = `π/10 - (3π)/10 - π/6 = (-11π)/30`