To simplify $\displaystyle 1+\frac{cos^2(\theta)}{sin^2(\theta)}$, find the common denominator and multiply the numerator accordingly.

$\displaystyle 1+\frac{cos^2(\theta)}{sin^2(\theta)}=\frac{sin^2(\theta)}{sin^2(\theta)}+\frac{cos^2(\theta)}{sin^2(\theta)}=\frac{sin^2(\theta)+cos^2(\theta)}{sin^2(\theta)}$

The numerator is an identity.

$\displaystyle sin^2(\theta)+cos^2(\theta)=1$

Substitute the identity and simplify.

$\displaystyle \frac{sin^2(\theta)+cos^2(\theta)}{sin^2(\theta)}= \frac{1}{sin^2(\theta)}= csc^2(\theta)$

Convert $\displaystyle \frac{sec^2(\theta)}{cot(\theta)}$ into its sines and cosines.

$\displaystyle \frac{sec^2(\theta)}{cot(\theta)}= \left(\frac{1}{cos(\theta)}\times\frac{1}{cos(\theta)}\right)\div \frac{cos(\theta)}{sin(\theta)}$

$\displaystyle \left(\frac{1}{cos(\theta)}\times\frac{1}{cos(\theta)}\right)\div \frac{cos(\theta)}{sin(\theta)}=\frac{1}{cos^2(\theta)}\times\frac{sin(\theta)}{cos(\theta)}= \frac{sin(\theta)}{cos^3(\theta)}$

$\displaystyle sin^3(x)+sin(x)\cdot cos^2(x)$

First factor out sine x.

$\displaystyle =(sin^2(x)+cos^2(x))\cdot sin(x)$

Notice that a Pythagorean Identity is present.

The identity needed for this problem is: 

$\displaystyle (sin^2(x)+cos^2(x))=1$

Using this identity the equation becomes,

$\displaystyle =(1)\cdot sin(x)$

$\displaystyle =sin(x)$.

To simplify, use the trigonometric identities $\displaystyle \frac{sin(x)}{cos(x)}=tan(x)$ and $\displaystyle \frac{1}{cot(x)}=tan(x)$ to rewrite both halves of the expression:

$\displaystyle tan(x)\times tan(x)$

Then combine using an exponent to simplify:

$\displaystyle tan^2(x)$

This expression is a trigonometric identity: $\displaystyle \frac{cos(x)}{sin(x)}=cot(x)$

Factor out 2 from the expression:

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

Then use the trigonometric identities $\displaystyle \frac{1}{csc(x)}=sin(x)$ and $\displaystyle \frac{1}{sec(x)}=cos(x)$ to rewrite the fractions:

$\displaystyle 2(sin^2(x)+cos^2(x))$

Finally, use the trigonometric identity $\displaystyle sin^2(x)+cos^2(x)=1$ to simplify:

$\displaystyle 2(1) = 2$

Factor out the common  $\displaystyle \frac{2}{tan(x)}$ from the expression:

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

Next, use the trigonometric identify $\displaystyle sin^2(x)+cos^2(x)=1$ to simplify:

$\displaystyle \frac{2}{tan(x)}(1)$

Then use the identify $\displaystyle \frac{1}{tan(x)}=cot(x)$ to simplify further:

$\displaystyle 2cot(x)$

To simplify the expression, separate the fraction into two parts:

$\displaystyle \frac{2tan^2x}{tan^2x}+\frac{1}{tan^2x}+tan^2x$

The $\displaystyle tan^2x$ terms in the first fraction cancel leaving you with:

$\displaystyle 2+\frac{1}{tan^2x}+tan^2x$

Then you can deal with the remaining fraction using the rule that $\displaystyle \frac{1}{tan^2x}=cot^2x$. This leaves:

$\displaystyle 2+cot^2x+tan^2x$

You can separate this into:

$\displaystyle 1+tan^2x+1+cot^2x$

And each half of this expression is now a trigonometric identity: $\displaystyle 1+tan^2x = sec^2x$ and $\displaystyle 1+cot^2x=csc^2x$. This gives you:

$\displaystyle sec^2x+csc^2x$