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

The general solution for sec(x)-cos(x)=sin(x) is x=2pin,x=pi+2pin,x= pi/4+pin

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

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

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

Solution steps

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

Subtract

sin (x)

from both sides

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

Express with sin, cos

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

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

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

Rewrite using trig identities

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

Factor

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

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

Solving each part separately

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

sin (x) = 0 : x = 2 πn, x = π + 2 πn

sin (x) - cos (x) = 0 : x = \frac{π}{4} + πn

Combine all the solutions

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

2 sin (x) = 3, 0 \leq x < 2 π

5 csc^{2} (x) + 2.5 cot (x) = 25

sec^{2} (x) + 0.5 tan (x) = 1

8 sin (x) = cos (x) - 3

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

What is the general solution for sec(x)-cos(x)=sin(x) ?
The general solution for sec(x)-cos(x)=sin(x) is x=2pin,x=pi+2pin,x= pi/4+pin