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

The general solution for tan(x)+sec(x)=2cos(x) is x= pi/6+2pin,x=(5pi)/6+2pin

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

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

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

Solution steps

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

Subtract

2 cos (x)

from both sides

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

Express with sin, cos

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

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

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

Rewrite using trig identities

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

Solve by substitution

sin (x) = \frac{1}{2}, sin (x) = - 1

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

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

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{3 π}{2} + 2 πn

Since the equation is undefined for:

\frac{3 π}{2} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

2 cos (x) + 1 = 0

sec (\frac{5 π}{6})

cos (7 5^{\circ})

cot (21 0^{\circ})

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

What is the general solution for tan(x)+sec(x)=2cos(x) ?
The general solution for tan(x)+sec(x)=2cos(x) is x= pi/6+2pin,x=(5pi)/6+2pin