What is the general solution for tan(x)cos(x)+cos(x)=0 ?

The general solution for tan(x)cos(x)+cos(x)=0 is x=(3pi)/4+pin

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tan(x)cos(x)+cos(x)=0

tan (x) cos (x) + cos (x) = 0

x = \frac{3 π}{4} + πn

Solution steps

tan (x) cos (x) + cos (x) = 0

Factor

tan (x) cos (x) + cos (x) : cos (x) (tan (x) + 1)

cos (x) (tan (x) + 1) = 0

Solving each part separately

cos (x) = 0 or tan (x) + 1 = 0

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

tan (x) + 1 = 0 : x = \frac{3 π}{4} + πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn, x = \frac{3 π}{4} + πn

Since the equation is undefined for:

\frac{π}{2} + 2 πn, \frac{3 π}{2} + 2 πn

x = \frac{3 π}{4} + πn

sin^{2} (x) (tan^{2} (x)) = 1 - sin^{2} (x)

2 sin^{2} (x) = 3 - 5 cos (x)

- 3 cos (2 θ) - 18 sin (θ) - 8 = - 5 sin (θ) - 6

cos (x) (2 cos (x) + 1) = 0

tan^{2} (x) + 4 tan (x) = 3

What is the general solution for tan(x)cos(x)+cos(x)=0 ?
The general solution for tan(x)cos(x)+cos(x)=0 is x=(3pi)/4+pin