The correct answer is:
cot θ = (cos θ)/(sin θ).
Explanation:
We know that the tangent ratio is opposite/adjacent (O/A).
The ratio for sine is opposite/hypotenuse (O/H), and the ratio for cosine is adjacent/hypotenuse (A/H).
If we divide these two,
[tex] \frac{\text{O}}{\text{H}}\div \frac{\text{A}}{\text{H}}
\\
\\=\frac{\text{O}}{\text{H}} \times \frac{\text{H}}{\text{A}}=\frac{\text{OH}}{\text{HA}}
\\
\\=\frac{\text{O}}{\text{A}} [/tex]
This is the same as the ratio for tangent, so we know that
tan θ = (sin θ)/(cos θ).
Since cot θ = 1/(tan θ), we now have
[tex] \cot \theta = \frac{1}{\frac{\sin \theta}{\cos \theta}}
\\
\\=1 \div \frac{\sin \theta}{\cos \theta}=1 \times \frac{\cos \theta}{\sin \theta}
\\
\\=\frac{\cos \theta}{\sin \theta} [/tex]
