What is the general solution for cos(x)(tan(x)csc(x))^2-1=2cos(x) ?

The general solution for cos(x)(tan(x)csc(x))^2-1=2cos(x) is x= pi/3+2pin,x=(5pi)/3+2pin

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cos(x)(tan(x)csc(x))^2-1=2cos(x)

cos (x) (tan (x) csc (x))^{2} - 1 = 2 cos (x)

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

Solution steps

cos (x) (tan (x) csc (x))^{2} - 1 = 2 cos (x)

Subtract

2 cos (x)

from both sides

tan^{2} (x) csc^{2} (x) cos (x) - 1 - 2 cos (x) = 0

Express with sin, cos

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

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

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

Solve by substitution

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

cos (x) = - 1 : x = π + 2 πn

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

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

Since the equation is undefined for:

π + 2 πn

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

sin (a) = \frac{4}{6}

cos (a) = 0.94

2 cos (2 x) - 1 = 0

2 cos^{2} (t) - cos (t) - 3 = 0

2 cos (4 θ + 20) = 0.8

What is the general solution for cos(x)(tan(x)csc(x))^2-1=2cos(x) ?
The general solution for cos(x)(tan(x)csc(x))^2-1=2cos(x) is x= pi/3+2pin,x=(5pi)/3+2pin