What is the general solution for sec(x)-(sin(x))/(cos(x))=cos(x) ?

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

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sec(x)-(sin(x))/(cos(x))=cos(x)

sec (x) - \frac{sin ( x )}{cos ( x )} = cos (x)

x = 2 πn, x = π + 2 πn

Solution steps

sec (x) - \frac{sin ( x )}{cos ( x )} = cos (x)

Subtract

cos (x)

from both sides

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

Simplify

sec (x) - \frac{sin ( x )}{cos ( x )} - cos (x) : \frac{sec ( x ) cos ( x ) - sin ( x ) - cos ^{2} ( x )}{cos ( x )}

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

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

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

Express with sin, cos

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

Simplify

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

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

Add

cos^{2} (x)

to both sides

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

Square both sides

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

Subtract

(cos^{2} (x))^{2}

from both sides

(1 - sin (x))^{2} - cos^{4} (x) = 0

Factor

(1 - sin (x))^{2} - cos^{4} (x) : (1 - sin (x) + cos^{2} (x)) (1 - sin (x) - cos^{2} (x))

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

Solving each part separately

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

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

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

Combine all the solutions

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

Verify solutions by plugging them into the original equation

x = 2 πn, x = π + 2 πn

- 10 sin^{2} (t) = 0

x, e^{y} cos (2 x) + y = 1

cos (θ) = 0.4, sec (θ)

- \frac{π}{2} sin (\frac{π x}{2}) = 0

cot (θ) = \frac{1 + cos ^{2} ( θ )}{2 sin ( θ ) cos ( θ )}

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