Question

Solve 3 tanx+cotx=5cosecx.

Answer 1

3tanx+cotx= 5cosecx

3sinx/cosx+cosx/sinx= 5/sinx
3sin2x+cos2x= 5cosx
3-3cos2x+cos2x= 5cosx
3-2cos2x=5cosx
2cos2x+5cosx-3= 0
2cos2x+6cosx-cosx-3=0
2cosx(cosx+3)+(-1)(cosx+3)=0
Cosx=-3,(1/2)
x=cos-1(-3),cos-1(1/2)
=60°,cos-1(-3)

Answer 2

Given Equation is 3 tan x + cot x = 5cosecx.

3(sinx/cosx) + (cosx/sinx) = 5(1/sinx)

On cross multiplication, we get

3sin^2x+cos^2x/cosxsinx = 5/sinx

3sin^2x + cos^2x = 5 * cosx sinx/sinx

3sin^2x + cos^2x = 5cosx

We know that sin^2x + cos^2x = 1 (or) sin^2x = 1-cos^2x.

3(1-cos^2x)+ cos^2x = 5cosx

3-3cos^2x+cos^2x=5cosx

3-2cos^2x=5cosx

2cos^2x+5cosx-3=0

2(cosx-1)(cosx+3)=0

2cosx-1 = 0 (or) cosx+3 = 0

cosx=1/2 (or)cosx = -3.

Hence cos x = 1/2

                   x = 60.


Hope this helps!