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

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

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

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

x = \frac{π}{2} + πn

Degrees

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

Solution steps

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

Subtract

sin^{2} (x)

from both sides

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

= 1 + cos (2 x)

1 + cos (2 x) = 0

Move

1

to the right side

1 + cos (2 x) = 0

Subtract

1

from both sides

1 + cos (2 x) - 1 = 0 - 1

Simplify

cos (2 x) = - 1

cos (2 x) = - 1

General solutions for

cos (2 x) = - 1

cos (x)

periodicity table with

2 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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

2 x = π + 2 πn

2 x = π + 2 πn

Solve

2 x = π + 2 πn : x = \frac{π}{2} + πn

2 x = π + 2 πn

Divide both sides by

2

2 x = π + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{π}{2} + \frac{2 πn}{2}

Simplify

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

x = \frac{π}{2} + πn

4 cos (2 x) = 2

cos (x) = \frac{5}{4}

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

3 cos^{2} (θ) + 7 cos (θ) - 9 = - 3

1 = 2 sin (x)

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