To equations are equivalent if and only if they have the same solution set.
Solve for both of the equations.
-2x = 10
x = -5 <-- Divide both sides by -2
-5x = 25
x = -5 <-- Divide both sides by -5
Both have the solution -5.
So, yes both are equivalent.
3.) 3sinθ = sinθ - 1
3sinθ - sinθ = -1
2sinθ = -1
sinθ = -1/2
θ = sin^-1(-1/2) = π + π/6, 2π - π/6 = 7π/6, 11π/6
4.) tan^2 θ = -3/2 sec θ
sin^2 θ / cos^2 θ = -3/2(1/cos θ)
sin^2 θ / cos θ = -3/2
1 - cos^2 θ = -3/2 cos θ
cos^2 θ - 3/2 cos θ - 1 = 0
2cos^2 θ - 3cos θ - 2 = 0
2cos^2 θ - 4cos θ + cos θ - 2 = 0
2cos θ (cos θ - 2) + 1(cos θ - 2) = 0
(2cos θ + 1) = 0 or (cos θ - 2) = 0
cos θ = -1/2 or cos θ = 2 [but cos θ can never be 2]
θ = cos^-1(-1/2) = π - π/3, π + π/3 = 2π/3, 4π/3
