# How do you prove that sinytany=cosy is not an identity by showing a counterexample?

Jan 29, 2017

It is not an identity. See explanation.

#### Explanation:

We have

$\sin y \tan y = \cos y$

This can be written as

$\sin y \cdot \sin \frac{y}{\cos} y = \cos y \text{ }$($\tan y = \sin \frac{y}{\cos} y$)

Solving this we get

${\sin}^{2} y = {\cos}^{2} y$

This will be true only when $\sin y = \cos y$.
For example, $\sin {45}^{\circ} = \cos 4 {5}^{\circ} = \frac{1}{\sqrt{2}}$ and so

$\sin {45}^{\circ} \tan {45}^{\circ} = \frac{1}{\sqrt{2}} \times 1 = \frac{1}{\sqrt{2}} = \cos {45}^{\circ}$

As a counterexample, we can say
$\sin {30}^{\circ} \tan {30}^{\circ} = \frac{1}{2} \times \frac{1}{\sqrt{3}} = \frac{1}{2 \sqrt{3}}$

but $\cos {30}^{\circ} = \frac{\sqrt{3}}{2}$

Hope that helps!