Question

find the general solution of
sinx+cos x+secx+cosecx=0
chapter trigonometry functions class 11

Answer 1

Given : Trigonometric equation, sinx + cosx + secx + cosecx = 0

To find : Find the general solution of given trigonometric equation.

solution : sinx + cosx + secx + cosecx = 0

⇒sinx + cosx + 1/cosx + 1/sinx = 0

⇒sinx + cosx + (sinx + cosx)/sinx . cosx = 0

⇒(sinx + cosx) [sinx cosx + 1]/sinx cosx = 0

⇒√2sin(x + π/4) {1/2(2sinx cosx) + 1}/sinx cosx = 0

⇒√2 sin(x + π/4) (sin2x + 2)/(2sinx cosx) = 0

⇒√2sin(x + π/4)(sin2x + 2)/sin2x = 0

from above,

sin(x + π/4) = 0 ⇒x = nπ - π/4

sin2x + 2 = 0 ⇒x = no solution

sin2x ≠ 0 ⇒x ≠ nπ

Therefore solution of given trigonometric equation is x = nπ - π/4 , where n is integral number.