Could someone please show me how to solve this
Prove
tan(-è) = -tanè
To prove the identity tan(-θ) = -tan(θ), where θ is any angle, we can use the definitions of tangent and the properties of trigonometric functions.
First, let's start by recalling the definition of the tangent function:
tan(θ) = sin(θ) / cos(θ)
Now, we want to prove that tan(-θ) = -tan(θ).
To begin the proof, let's calculate tan(-θ) using the definition of the tangent function:
tan(-θ) = sin(-θ) / cos(-θ)
Next, we'll use two properties of trigonometric functions:
1. sin(-θ) = -sin(θ) (property of sine function)
2. cos(-θ) = cos(θ) (property of cosine function)
Substituting these values into our expression, we get:
tan(-θ) = -sin(θ) / cos(θ)
Now, let's simplify this expression by multiplying the numerator and denominator by -1:
tan(-θ) = (-1)(-sin(θ)) / (-1)(cos(θ))
tan(-θ) = sin(θ) / cos(θ)
Recall the definition of the tangent function:
tan(θ) = sin(θ) / cos(θ)
We can observe that tan(-θ) is equal to tan(θ). Since we multiplied both the numerator and denominator by -1, the sign of the tangent function changed.
Therefore, we have proven that tan(-θ) = -tan(θ), which was the original identity we wanted to prove.