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

The general solution for sec(2x)+tan(2x)=2 is x=0.32175…+pin

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

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

x = 0.32175 \dots + πn

Solution steps

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

Subtract

2

from both sides

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

Express with sin, cos

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

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

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

Rewrite using trig identities

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

Factor

- cos^{2} (x) + 3 sin^{2} (x) + 2 cos (x) sin (x) : (3 sin (x) - cos (x)) (sin (x) + cos (x))

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

Solving each part separately

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

3 sin (x) - cos (x) = 0 : x = arctan (\frac{1}{3}) + πn

sin (x) + cos (x) = 0 : x = \frac{3 π}{4} + πn

Combine all the solutions

x = arctan (\frac{1}{3}) + πn, x = \frac{3 π}{4} + πn

Since the equation is undefined for:

\frac{3 π}{4} + πn

x = arctan (\frac{1}{3}) + πn

Show solutions in decimal form

x = 0.32175 \dots + πn

sec (2 x) + tan (2 x) = 9

2 + 3 cos^{2} (x) = sin^{2} (x) + 4 cos (x)

sin (x) - 3 cos (x) = 0

tan (4 x) - tan (2 x) = 0

t, sin (t) + 2 sin (2 t) = 0

What is the general solution for sec(2x)+tan(2x)=2 ?
The general solution for sec(2x)+tan(2x)=2 is x=0.32175…+pin