Question

simplify $\frac{secx^{2} }{cotx^{2}+1}$

Answer 1

Answer: $tan(x)^{2}$

Step-by-step explanation:

We will use the trigonometric identities to solve this problem:

$\frac{sec(x)^{2}}{cot(x)^{2}+1}$ (1)

Let's begin by the following trigonometric identity:

$sec(x)^{2}=tan(x)^{2}+1$ (2)

An substitute it in (1):

$\frac{tan(x)^{2}+1}{cot(x)^{2}+1}$ (3)

Then, taking into account $tan(x)^{2}=\frac{sin(x)^{2}}{cos(x)^{2}}$ and $cot(x)^{2}=\frac{cos(x)^{2}}{sin(x)^{2}}$, we rewrite (3):

$\frac{\frac{sin(x)^{2}}{cos(x)^{2}}+1}{\frac{cos(x)^{2}}{sin(x)^{2}}+1}$ (4)

$\frac{\frac{sin(x)^{2}+cos(x)^{2}}{cos(x)^{2}}}{\frac{cos(x)^{2}+sin(x)^{2}}{sin(x)^{2}}}$ (5)

Then, applying the trigonometric identity $sin(x)^{2}+cos(x)^{2}=1$

$\frac{1}{cos(x)^{2}}}{\frac{1}{sin(x)^{2}}}$ (6)

Finally

$\frac{sin(x)^{2}}{cos(x)^{2}}}=tan(x)^{2}$ (7)