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

The general solution for tan(x)=sin(x)cos(x) is x=2pin,x=pi+2pin

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tan(x)=sin(x)cos(x)

tan (x) = sin (x) cos (x)

x = 2 πn, x = π + 2 πn

Solution steps

tan (x) = sin (x) cos (x)

Subtract

sin (x) cos (x)

from both sides

tan (x) - sin (x) cos (x) = 0

Express with sin, cos

\frac{sin ( x ) - cos ^{2} ( x ) sin ( x )}{cos ( x )} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

sin (x) - cos^{2} (x) sin (x) = 0

Factor

sin (x) - cos^{2} (x) sin (x) : - sin (x) (cos (x) + 1) (cos (x) - 1)

- sin (x) (cos (x) + 1) (cos (x) - 1) = 0

Solving each part separately

sin (x) = 0 or cos (x) + 1 = 0 or cos (x) - 1 = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

cos (x) + 1 = 0 : x = π + 2 πn

cos (x) - 1 = 0 : x = 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn

3 sin (θ) - cos (θ) = 1

csc (y) = 2

16 sin^{2} (θ) - 1 = 0

8 sin^{2} (x) + 16 sin (x) + 8 = 0

tan (θ) = \frac{2 3}{3} sin (θ), 0 \leq θ < 2 π

What is the general solution for tan(x)=sin(x)cos(x) ?
The general solution for tan(x)=sin(x)cos(x) is x=2pin,x=pi+2pin