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

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

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sin(x)+cos(x)=sec(x)

sin (x) + cos (x) = sec (x)

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

Solution steps

sin (x) + cos (x) = sec (x)

Subtract

sec (x)

from both sides

sin (x) + cos (x) - sec (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

y, x = tan (y^{2})

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

sin (x) = - 0.2761

3 arcsin (x) = \frac{π}{2}, x

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

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