# How do you prove  1 + 1/(tan^2x) = 1/(sin^2x)?

Dec 12, 2015

I tried changing $\tan$ into $\sin \mathmr{and} \cos$:

#### Explanation:

You can change the tangent as:
$\tan \left(x\right) = \sin \frac{x}{\cos} \left(x\right)$ and write it as:
$1 + \frac{1}{{\sin}^{2} \frac{x}{\cos} ^ 2 \left(x\right)} = \frac{1}{\sin} ^ 2 \left(x\right)$
rearranging the left side:
$1 + {\cos}^{2} \frac{x}{\sin} ^ 2 \left(x\right) = \frac{1}{\sin} ^ 2 \left(x\right)$
$\frac{{\sin}^{2} \left(x\right) + {\cos}^{2} \left(x\right)}{\sin} ^ 2 \left(x\right) = \frac{1}{\sin} ^ 2 \left(x\right)$
but: ${\sin}^{2} \left(x\right) + {\cos}^{2} \left(x\right) = 1$
so:
$\frac{1}{\sin} ^ 2 \left(x\right) = \frac{1}{\sin} ^ 2 \left(x\right)$