What is the general solution for csc^2(x)+3cot^2(x)=1 ?

The general solution for csc^2(x)+3cot^2(x)=1 is x= pi/2+pin

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csc^2(x)+3cot^2(x)=1

csc^{2} (x) + 3 cot^{2} (x) = 1

x = \frac{π}{2} + πn

Degrees

x = 9 0^{\circ} + 18 0^{\circ} n

Solution steps

csc^{2} (x) + 3 cot^{2} (x) = 1

Subtract

1

from both sides

csc^{2} (x) + 3 cot^{2} (x) - 1 = 0

Rewrite using trig identities

- 1 + csc^{2} (x) + 3 cot^{2} (x)

Use the Pythagorean identity:

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

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

= 3 cot^{2} (x) + cot^{2} (x)

Simplify

= 4 cot^{2} (x)

4 cot^{2} (x) = 0

Divide both sides by

4

4 cot^{2} (x) = 0

Divide both sides by

4

4 cot^{2} (x) = 0

Divide both sides by

4

\frac{4 cot ^{2} ( x )}{4} = \frac{0}{4}

Simplify

cot^{2} (x) = 0

cot^{2} (x) = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

cot (x) = 0

General solutions for

cot (x) = 0

cot (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

2 sin^{2} (θ) - sin (θ) - 3 = 0

4 sin (x) cos (x) + 1 = 0

4 sin (x) = 3 csc (x)

sin (2 θ) = \frac{3}{2}, 0 \leq θ < 2 π

- sin (4 x) = 0

What is the general solution for csc^2(x)+3cot^2(x)=1 ?
The general solution for csc^2(x)+3cot^2(x)=1 is x= pi/2+pin