What is the general solution for cot^2(x)+csc(x)=1 ?

The general solution for cot^2(x)+csc(x)=1 is x= pi/2+2pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin

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cot^2(x)+csc(x)=1

cot^{2} (x) + csc (x) = 1

x = \frac{π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solution steps

cot^{2} (x) + csc (x) = 1

Subtract

1

from both sides

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

Rewrite using trig identities

- 2 + csc (x) + csc^{2} (x) = 0

Solve by substitution

csc (x) = 1, csc (x) = - 2

csc (x) = 1 : x = \frac{π}{2} + 2 πn

csc (x) = - 2 : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{2} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

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

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

cos (2 x) - 2 cos (x) = - 1

cos (x) = \frac{3}{2}

- 3 sin (3 x) = 0

What is the general solution for cot^2(x)+csc(x)=1 ?
The general solution for cot^2(x)+csc(x)=1 is x= pi/2+2pin,x=(7pi)/6+2pin,x=(11pi)/6+2pin