Could someone please help me answer this
Prove tan(-theta) = -tan(theta)
To prove that tan(-θ) = -tan(θ), we need to use the definitions and properties of the tangent function.
First, let's recall the definition of the tangent function:
tan(θ) = sin(θ) / cos(θ)
Now, let's consider tan(-θ):
tan(-θ) = sin(-θ) / cos(-θ)
To simplify this, we can use the properties of sine and cosine:
sin(-θ) = -sin(θ) ... (1)
cos(-θ) = cos(θ) ... (2)
Equation (1) is derived from the fact that sine is an odd function, meaning sin(-x) = -sin(x).
Equation (2) is derived from the fact that cosine is an even function, meaning cos(-x) = cos(x).
Now substitute equations (1) and (2) back into the expression for tan(-θ):
tan(-θ) = -sin(θ) / cos(θ) (substituting sin(-θ) and cos(-θ))
We can also simplify this expression further:
tan(-θ) = -(sin(θ) / cos(θ)) (multiplying numerator and denominator by -1)
Since -(sin(θ) / cos(θ)) is equal to -tan(θ), we've proven that tan(-θ) = -tan(θ).
Therefore, tan(-θ) = -tan(θ) is true.