What is the general solution for 2csc^2(x)=5-5cot(x) ?

The general solution for 2csc^2(x)=5-5cot(x) is x=1.10714…+pin,x=2.81984…+pin

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2csc^2(x)=5-5cot(x)

2 csc^{2} (x) = 5 - 5 cot (x)

x = 1.10714 \dots + πn, x = 2.81984 \dots + πn

Solution steps

2 csc^{2} (x) = 5 - 5 cot (x)

Subtract

5 - 5 cot (x)

from both sides

2 csc^{2} (x) - 5 + 5 cot (x) = 0

Rewrite using trig identities

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

Solve by substitution

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

cot (x) = \frac{1}{2} : x = arccot (\frac{1}{2}) + πn

cot (x) = - 3 : x = arccot (- 3) + πn

Combine all the solutions

x = arccot (\frac{1}{2}) + πn, x = arccot (- 3) + πn

Show solutions in decimal form

x = 1.10714 \dots + πn, x = 2.81984 \dots + πn

sin (x + 0.5) = 0.92, 0 \leq x \leq 2 π

cos (4 x) = 0.8 - 0.6

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

\frac{18 cos ( 2 θ ) + 2 + cos ( θ )}{2} = 0

0 = 3 sin (x - \frac{1}{2} π) + 2

What is the general solution for 2csc^2(x)=5-5cot(x) ?
The general solution for 2csc^2(x)=5-5cot(x) is x=1.10714…+pin,x=2.81984…+pin