What is the general solution for tan(x)+3cot(x)=5sec(x) ?

The general solution for tan(x)+3cot(x)=5sec(x) is x= pi/6+2pin,x=(5pi)/6+2pin

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tan(x)+3cot(x)=5sec(x)

tan (x) + 3 cot (x) = 5 sec (x)

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

Solution steps

tan (x) + 3 cot (x) = 5 sec (x)

Subtract

5 sec (x)

from both sides

tan (x) + 3 cot (x) - 5 sec (x) = 0

Express with sin, cos

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

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

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

Rewrite using trig identities

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

Solve by substitution

sin (x) = - 3, sin (x) = \frac{1}{2}

sin (x) = - 3 :

No Solution

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

Combine all the solutions

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

6 sin (θ) + 3 = 0

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

sin (3 θ) = - \frac{1}{2}

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

sec^{2} (x) + 5 tan (x) = 7

What is the general solution for tan(x)+3cot(x)=5sec(x) ?
The general solution for tan(x)+3cot(x)=5sec(x) is x= pi/6+2pin,x=(5pi)/6+2pin