# Question #5a859

Aug 14, 2016

We will start from the left hand side and show that it equals the right hand side. To do so, we will use the following identities:

• $\tan \left(x\right) = \sin \frac{x}{\cos} \left(x\right)$
• $\sec \left(x\right) = \frac{1}{\cos} \left(x\right)$
• ${\sec}^{2} \left(x\right) - 1 = {\tan}^{2} \left(x\right)$

Note that the third identity may be derived from ${\sin}^{2} \left(x\right) + {\cos}^{2} \left(x\right) = 1$ by dividing each side by ${\cos}^{2} \left(x\right)$ and then subtracting $1$ from each side.

Proceeding:

$L H S = {\tan}^{2} \left(u\right) - {\sin}^{2} \left(u\right)$

$= {\left(\sin \frac{u}{\cos} \left(u\right)\right)}^{2} - {\sin}^{2} \left(u\right)$

$= {\sin}^{2} \frac{u}{\cos} ^ 2 \left(u\right) - {\sin}^{2} \left(u\right)$

$= {\sin}^{2} \left(u\right) \left(\frac{1}{\cos} ^ 2 \left(u\right) - 1\right)$

$= {\sin}^{2} \left(u\right) \left({\sec}^{2} \left(u\right) - 1\right)$

$= {\sin}^{2} \left(u\right) {\tan}^{2} \left(u\right) = R H S$