What is the general solution for 4cot^2(x)=cos^2(x)+sin^2(x)-1 ?

The general solution for 4cot^2(x)=cos^2(x)+sin^2(x)-1 is x= pi/2+pin

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4cot^2(x)=cos^2(x)+sin^2(x)-1

Solution

4 cot^{2} (x) = cos^{2} (x) + sin^{2} (x) - 1

Solution

x = \frac{π}{2} + πn

Degrees

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

Solution steps

4 cot^{2} (x) = cos^{2} (x) + sin^{2} (x) - 1

Subtract

cos^{2} (x) + sin^{2} (x) - 1

from both sides

4 cot^{2} (x) - cos^{2} (x) - sin^{2} (x) + 1 = 0

Rewrite using trig identities

4 cot^{2} (x) - cos^{2} (x) - sin^{2} (x) + 1

Use the Pythagorean identity:

cos^{2} (x) + sin^{2} (x) = 1

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

= 1 + 4 cot^{2} (x) - 1

1 + 4 cot^{2} (x) - 1 = 0

Move

1

to the right side

1 + 4 cot^{2} (x) - 1 = 0

Subtract

1

from both sides

1 + 4 cot^{2} (x) - 1 - 1 = 0 - 1

Simplify

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

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

Move

1

to the right side

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

Add

1

to both sides

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

Simplify

4 cot^{2} (x) = 0

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

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for 4cot^2(x)=cos^2(x)+sin^2(x)-1 ?
The general solution for 4cot^2(x)=cos^2(x)+sin^2(x)-1 is x= pi/2+pin