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

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

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

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

x = πn

Degrees

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

Solution steps

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

Subtract

sin^{2} (x) + 1

from both sides

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

Rewrite using trig identities

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

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

Add

1

to 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 = 0 + 2 πn

2 x = 0 + 2 πn

Solve

2 x = 0 + 2 πn : x = πn

2 x = 0 + 2 πn

0 + 2 πn = 2 πn

2 x = 2 πn

Divide both sides by

2

2 x = 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{2 πn}{2}

Simplify

x = πn

x = πn

x = πn

4 sin (θ) cos^{2} (θ) = 3 sin (θ)

sin (x) = \frac{7}{25}

cos (x) + 9 sin^{2} (x) = 1

- \frac{2}{3} cos (x) + \frac{2}{3} cos (2 x) = 0

sin (θ) - cos (θ) = 0

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