What is the general solution for tan(x)+sec(x)=2 ?

The general solution for tan(x)+sec(x)=2 is x=0.64350…+2pin

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tan(x)+sec(x)=2

tan (x) + sec (x) = 2

x = 0.64350 \dots + 2 πn

Solution steps

tan (x) + sec (x) = 2

Subtract

2

from both sides

tan (x) + sec (x) - 2 = 0

Express with sin, cos

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

Simplify

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

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

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

sin (x) + 1 - 2 cos (x) = 0

Add

2 cos (x)

to both sides

sin (x) + 1 = 2 cos (x)

Square both sides

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

Subtract

(2 cos (x))^{2}

from both sides

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

Rewrite using trig identities

- 3 + 2 sin (x) + 5 sin^{2} (x) = 0

Solve by substitution

sin (x) = \frac{3}{5}, sin (x) = - 1

sin (x) = \frac{3}{5} : x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn

sin (x) = - 1 : x = \frac{3 π}{2} + 2 πn

Combine all the solutions

x = arcsin (\frac{3}{5}) + 2 πn, x = π - arcsin (\frac{3}{5}) + 2 πn, x = \frac{3 π}{2} + 2 πn

Verify solutions by plugging them into the original equation

x = arcsin (\frac{3}{5}) + 2 πn

Show solutions in decimal form

x = 0.64350 \dots + 2 πn

sin (x) = \frac{4}{5}, \frac{π}{2} \leq x \leq π

2 - 2 cosh (t - 2) = 0

cos (x) = \frac{1}{37}

\frac{sin ( 2 x )}{cos ( x )} = 4 + sin (2 x) \cdot tan (x)

sin (x) = 0.2174

What is the general solution for tan(x)+sec(x)=2 ?
The general solution for tan(x)+sec(x)=2 is x=0.64350…+2pin