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

The general solution for sin^4(x)+cos^4(x)=1 is x=2pin,x=pi+2pin,x= pi/2+2pin,x=(3pi)/2+2pin

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

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

x = 2 πn, x = π + 2 πn, x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Solution steps

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

Subtract

1

from both sides

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

Apply exponent rule:

a^{b} = a^{2} a^{b - 2}

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

Rewrite using trig identities

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

Solve by substitution

cos (x) = 1, cos (x) = - 1, cos (x) = 0

cos (x) = 1 : x = 2 πn

cos (x) = - 1 : x = π + 2 πn

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

Combine all the solutions

x = 2 πn, x = π + 2 πn, x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

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

tan (θ) = \frac{8}{15}

tan (x) = - 6

sin (7 x) + sin (x) = 0

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

What is the general solution for sin^4(x)+cos^4(x)=1 ?
The general solution for sin^4(x)+cos^4(x)=1 is x=2pin,x=pi+2pin,x= pi/2+2pin,x=(3pi)/2+2pin