tan^2(x)sin(x)=sin(x),0<= x<= 2pi

Solution

tan^{2} (x) sin (x) = sin (x), 0 \leq x \leq 2 π

x = 0, x = π, x = 2 π, x = \frac{3 π}{4}, x = \frac{7 π}{4}, x = \frac{π}{4}, x = \frac{5 π}{4}

Solution steps

tan^{2} (x) sin (x) = sin (x), 0 \leq x \leq 2 π

Subtract

sin (x)

from both sides

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

Factor

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

sin (x) (tan (x) + 1) (tan (x) - 1) = 0

Solving each part separately

sin (x) = 0 or tan (x) + 1 = 0 or tan (x) - 1 = 0

sin (x) = 0, 0 \leq x \leq 2 π : x = 0, x = π, x = 2 π

tan (x) + 1 = 0, 0 \leq x \leq 2 π : x = \frac{3 π}{4}, x = \frac{7 π}{4}

tan (x) - 1 = 0, 0 \leq x \leq 2 π : x = \frac{π}{4}, x = \frac{5 π}{4}

Combine all the solutions

x = 0, x = π, x = 2 π, x = \frac{3 π}{4}, x = \frac{7 π}{4}, x = \frac{π}{4}, x = \frac{5 π}{4}

- 3 sin (t) = 0

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

3 cos^{2} (θ) + 7 cos (θ) - 8 = 0

2 cos (2 a) = 2, 0 \leq θ \leq 2 π

t, y^{4} - 81 y = sin (t)