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:

\(\displaystyle sec^2x+csc^2x\)