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

The general solution for (cos(x)-sin(x))/(cos(x))=tan(x)-1 is x= pi/4+pin

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

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

x = \frac{π}{4} + πn

Solution steps

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

Subtract

tan (x) - 1

from both sides

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

Simplify

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

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

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

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

Rewrite using trig identities

2 - 2 tan (x) = 0

Move

2

to the right side

- 2 tan (x) = - 2

Divide both sides by

- 2

tan (x) = 1

General solutions for

tan (x) = 1

x = \frac{π}{4} + πn

cos (φ) = \frac{1}{4}

- 2 cos (x) = sin (x)

sin (x) = - 0.9781

tan (x) = 3.73

tan (θ) = - \frac{2}{5}, sin (θ) > 0

What is the general solution for (cos(x)-sin(x))/(cos(x))=tan(x)-1 ?
The general solution for (cos(x)-sin(x))/(cos(x))=tan(x)-1 is x= pi/4+pin