To prove that tan(θ) cot(θ) = 1, we start by recalling the definitions of the tangent and cotangent functions:
- tan(θ) = sin(θ) / cos(θ)
- cot(θ) = cos(θ) / sin(θ)
Now, let us substitute these definitions into the expression tan(θ) cot(θ):
tan(θ) cot(θ) = (sin(θ) / cos(θ)) * (cos(θ) / sin(θ))
When we multiply these fractions, we notice that the sin(θ) in the numerator of tan(θ) and the sin(θ) in the denominator of cot(θ) will cancel each other out, as will the cos(θ) in the numerator of cot(θ) and the cos(θ) in the denominator of tan(θ):
tan(θ) cot(θ) = (sin(θ) * cos(θ)) / (cos(θ) * sin(θ)) = 1
Thus, we have proven that:
tan(θ) cot(θ) = 1
This identity holds for all values of θ where tangent and cotangent are defined.