cos^4(x)-sin^4(x)= 1/2

Solution

cos^{4} (x) - sin^{4} (x) = \frac{1}{2}

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Solution steps

cos^{4} (x) - sin^{4} (x) = \frac{1}{2}

Subtract

\frac{1}{2}

from both sides

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

Simplify

cos^{4} (x) - sin^{4} (x) - \frac{1}{2} : \frac{2 cos ^{4} ( x ) - 2 sin ^{4} ( x ) - 1}{2}

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Apply exponent rule:

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

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

Rewrite using trig identities

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

Solve by substitution

sin (x) = \frac{1}{2}, sin (x) = - \frac{1}{2}

sin (x) = \frac{1}{2} : x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn

sin (x) = - \frac{1}{2} : x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{5 π}{6} + 2 πn, x = \frac{7 π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sin (x) + cos (x) = tan (x) + 1

cos (x) = \frac{1}{4}, 0 \leq x \leq 2 π

tan^{2} (x) + 1 = a^{2}

2 = 2 cos (t)

50 sin (4 5^{\circ}) = 600 sin (θ)