What is the general solution for (cos(x))/(1+sin(x))+tan(x)=cos(x) ?

The general solution for (cos(x))/(1+sin(x))+tan(x)=cos(x) is x=2pin,x=pi+2pin

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(cos(x))/(1+sin(x))+tan(x)=cos(x)

\frac{cos ( x )}{1 + sin ( x )} + tan (x) = cos (x)

x = 2 πn, x = π + 2 πn

Solution steps

\frac{cos ( x )}{1 + sin ( x )} + tan (x) = cos (x)

Subtract

cos (x)

from both sides

\frac{tan ( x ) + tan ( x ) sin ( x ) - cos ( x ) sin ( x )}{1 + sin ( x )} = 0

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

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

Express with sin, cos

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

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

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

Factor

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

sin (x) (1 + sin (x) - cos^{2} (x)) = 0

Solving each part separately

sin (x) = 0 or 1 + sin (x) - cos^{2} (x) = 0

sin (x) = 0 : x = 2 πn, x = π + 2 πn

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

Combine all the solutions

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

Since the equation is undefined for:

\frac{3 π}{2} + 2 πn

x = 2 πn, x = π + 2 πn

sin (2 x) = 2 cos^{2} (x), 0 \leq x \leq 2 π

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

sin (2 x) = - 1 sin (x)

cos (x) = \frac{1}{6}

2 cos^{2} (θ) + 9 cos (θ) + 4 = 0

What is the general solution for (cos(x))/(1+sin(x))+tan(x)=cos(x) ?
The general solution for (cos(x))/(1+sin(x))+tan(x)=cos(x) is x=2pin,x=pi+2pin