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

Solution

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

x = - 0.66623 \dots + 2 πn, x = π + 0.66623 \dots + 2 πn, x = 0.66623 \dots + 2 πn, x = π - 0.66623 \dots + 2 πn

Solution steps

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

Subtract

1

from both sides

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

Rewrite using trig identities

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

Factor

- cos^{2} (x) + tan^{2} (x) : (tan (x) + cos (x)) (tan (x) - cos (x))

(tan (x) + cos (x)) (tan (x) - cos (x)) = 0

Solving each part separately

tan (x) + cos (x) = 0 or tan (x) - cos (x) = 0

tan (x) + cos (x) = 0 : x = arcsin (- \frac{- 1 + 5}{2}) + 2 πn, x = π + arcsin (\frac{- 1 + 5}{2}) + 2 πn

tan (x) - cos (x) = 0 : x = arcsin (\frac{- 1 + 5}{2}) + 2 πn, x = π - arcsin (\frac{- 1 + 5}{2}) + 2 πn

Combine all the solutions

x = arcsin (- \frac{- 1 + 5}{2}) + 2 πn, x = π + arcsin (\frac{- 1 + 5}{2}) + 2 πn, x = arcsin (\frac{- 1 + 5}{2}) + 2 πn, x = π - arcsin (\frac{- 1 + 5}{2}) + 2 πn

Show solutions in decimal form

x = - 0.66623 \dots + 2 πn, x = π + 0.66623 \dots + 2 πn, x = 0.66623 \dots + 2 πn, x = π - 0.66623 \dots + 2 πn

cos (6 x) = sin (x - 1)

tan (θ) = \frac{72}{65}

cos (x) + cos (\frac{x}{2}) + 1 = 0

sin (x - 1) = sin (x)

sin (θ) = \frac{1.5}{2}