What is the general solution for sec(x)-tan(x)=1 ?

The general solution for sec(x)-tan(x)=1 is x=2pin

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sec(x)-tan(x)=1

sec (x) - tan (x) = 1

x = 2 πn

Solution steps

sec (x) - tan (x) = 1

Subtract

1

from both sides

sec (x) - tan (x) - 1 = 0

Express with sin, cos

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

Simplify

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

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

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

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

Add

cos (x)

to both sides

1 - sin (x) = cos (x)

Square both sides

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

Subtract

cos^{2} (x)

from both sides

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

Rewrite using trig identities

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

Solve by substitution

sin (x) = 1, sin (x) = 0

sin (x) = 1 : x = \frac{π}{2} + 2 πn

sin (x) = 0 : 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

7 cos^{2} (θ) + 6 cos (θ) - 12 = 7 cos (θ) - 6

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

tan^{2} (x) - 3 sec (x) + 3 = 0

cos (5 t) cos (2 t) = - sin (5 t) sin (2 t)

x, cos (x) = sin (y)

What is the general solution for sec(x)-tan(x)=1 ?
The general solution for sec(x)-tan(x)=1 is x=2pin