Answer:
sinΘ
Explanation:
Assuming 0 was being used as variable Θ (theta):
For the numerator: sin(90° - Θ) = cosΘ (and conversely, cos(90° - Θ) = sinΘ because sin and cos are complementary functions). Recall the trigonometric identities secΘ = 1/cosΘ and sin(-Θ) = -sinΘ (this is because cos is an even function whilst sin is an odd function). This means that by simplifying the numerator, you will have cosΘ • (1/-cosΘ) • (-sinΘ) = sinΘ.
For the denominator: Recall that the periodicity of sin and cos functions is 360° (or 2π). This means that either function will have completed a full rotation upon the addition or subtraction of 360° (for example: sin(x + 360°) = sinx or cos(x + 360°) = cosx). Given that the angle for the sin function in this problem is 180°, this means that only half of a rotation is complete, and therefore the quadrantal sign changes to its inverse (from + to - in this case). sin(180° + Θ) = -sinΘ. For tan and cot functions, the periodicity is 180° (or π). Since we're using 360° for this particular problem, there are 2 full rotations (again, this is only for tan and cot), creating a negative angle. Therefore, cot(360° - Θ) = -cotΘ. Now use the identity cscΘ = 1/sinΘ. This means that csc(90° + Θ) = 1/sin(90° + Θ). As established before, we can determine that sin(90° + Θ) = cosΘ (this is due to the conjugate property, more specifically sin(90° ± Θ) = cosΘ). This implies that csc(90° + Θ) = 1/cosΘ = secΘ. Now multiply the factors in the denominator: -sinΘ • (-cotΘ) • secΘ = -sinΘ • (-cosΘ/sinΘ) • 1/cosΘ = 1.
Now to finish simplifying the fraction: -sinΘ/1 = sinΘ.
If anyone notices an error in my explanation, please correct me, and thanks in advance.