Question

How do you prove $ \sin \theta \cot \theta =\cos \theta $ ?

Answer 1

Hint: We first try to find the respective ratios of the multiplication $ \sin \theta \cot \theta $ . We find their respective values according to the sides of a right-angle triangle. We use the relations $ \sin \theta =\dfrac{height}{hypotenuse} $ and $ \cot \theta =\dfrac{base}{height} $ to multiply them. Then from the final ration we find the solution.

Complete step-by-step answer:
The given trigonometric expression is the multiplication of two ratios $ \sin \theta $ and $ \cot \theta $ .
We have their respective values according to the sides of a right-angle triangle. We use those relations to find the value of $ \sin \theta \cot \theta $ .
According to a right-angle triangle the value of $ \sin \theta $ will be considered as the ratio of the length of the height to the hypotenuse with respect to a certain angle.
So, $ \sin \theta =\dfrac{height}{hypotenuse} $ .
And according to the same right-angle triangle the value of $ \cot \theta $ will be considered as the ratio of the length of the base to the height with respect to the same angle.
So, $ \cot \theta =\dfrac{base}{height} $ .
The multiplied form of the term $ \sin \theta \cot \theta $ gives $ \sin \theta \cot \theta =\dfrac{height}{hypotenuse}\times \dfrac{base}{height}=\dfrac{base}{hypotenuse} $ .
The ratio of $ \dfrac{base}{hypotenuse} $ is defined as $ \dfrac{base}{hypotenuse}=\cos \theta $ .
Therefore, $ \sin \theta \cot \theta =\cos \theta $ .
So, the correct answer is “ $ \sin \theta \cot \theta =\cos \theta $ ”.

Note: We can also use the direct ratio relation to find the solution. We know that $ \cot \theta $ can be broken into a ratio of two other trigonometric expressions which are $ \cos \theta $ and $ \sin \theta $ . We know that $ \cot \theta =\dfrac{\cos \theta }{\sin \theta } $ . Now we multiply $ \sin \theta $ on both sides of the equation and get
\[\begin{align}
  & \cot \theta \times \sin \theta=\dfrac{\cos \theta }{\sin \theta }\times \sin \theta \\
 & \Rightarrow \sin \theta \cot \theta =\cos \theta \\
\end{align}\].
Thus, verified the relation $ \sin \theta \cot \theta =\cos \theta $ .