Because  \(\displaystyle \sin ^{2} x + \cos ^{2} x = 1\), 

therefore:

\(\displaystyle \small \frac{\sin ^2 x + \cos^2 x}{\sin x} = \frac{1}{\sin x}\)

By definition of cosecant,

\(\displaystyle \small \frac{1}{\sin x} = \csc x\)

There are several ways to work this problem, but all of them use the second Pythagorean trig identity, \(\displaystyle \small \tan ^2 x + 1 = \sec^2 x\).

You can use this identity to substitute for parts of the expression. Here are two examples.

Method 1:

Substituting \(\displaystyle -(\tan^2 x +1)\) for \(\displaystyle \small -\sec^2 x\), we get

\(\displaystyle \tan ^2 x +1 - (\tan ^2 x +1)\), which equals zero.

Method 2:

Substituting \(\displaystyle \sec^2 x\) for \(\displaystyle \tan^2 x +1\), we get

\(\displaystyle \sec^2 x-\sec^2 x\), which equals zero.

Regardless of which substitution you choose, the answer is the same.

The first step to simplifying is to remember an important trig identity.

\(\displaystyle \sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1\)

If we rewrite it to look like the denominator, it is.

\(\displaystyle \sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1\)

Now we can substitute this in the denominator.

\(\displaystyle \frac{\sin^{3}{\left (x \right )} + \cos{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}\)

Now write each term separately.

\(\displaystyle \frac{1}{2} \sin{\left (x \right )} + \frac{\cos{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}\)

Remember the following identities.

\(\displaystyle \\\csc{\left (x \right )} = \frac{1}{\sin{\left (x \right )}} \\\\\sec{\left (x \right )} = \frac{1}{\cos{\left (x \right )}} \\\\\tan{\left (x \right )} = \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \\\\\cot{\left (x \right )} = \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}\)

Now simplify, and combine each term.

\(\displaystyle \\\frac{1}{2} \sin{\left (x \right )} + \frac{1}{2} \cos{\left (x \right )} \csc^{2}{\left (x \right )} \\\\\frac{1}{2} \left(\sin{\left (x \right )} + \cos{\left (x \right )} \csc^{2}{\left (x \right )}\right)\)

The first step to simplifying is to remember an important trig identity.

\(\displaystyle \sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1\)

If we rewrite it to look like the denominator, it is.

\(\displaystyle \sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1\)

Now we can substitute this in the denominator.

\(\displaystyle \frac{\sin^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}\)

Now write each term separately.

\(\displaystyle \frac{1}{2} + \frac{\cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}\)

Remember the following identities.

\(\displaystyle \\\csc{\left (x \right )} = \frac{1}{\sin{\left (x \right )}} \\\\\sec{\left (x \right )} = \frac{1}{\cos{\left (x \right )}} \\\\\tan{\left (x \right )} = \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \\\\\cot{\left (x \right )} = \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}\)

Now simplify, and combine each term.

\(\displaystyle \\\frac{1}{2} \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + \frac{1}{2} \\\\\frac{1}{2} \left(\cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)\)

The first step to simplifying is to remember an important trig identity.

\(\displaystyle \sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1\)

If we rewrite it to look like the denominator, it is.

\(\displaystyle \sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1\)

Now we can substitute this in the denominator.

\(\displaystyle \frac{\sin^{2}{\left (x \right )} + \cos{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}\)

Now write each term separately.

\(\displaystyle \frac{1}{2} + \frac{\cos{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}\)

Remember the following identities.

\(\displaystyle \\\csc{\left (x \right )} = \frac{1}{\sin{\left (x \right )}} \\\\\sec{\left (x \right )} = \frac{1}{\cos{\left (x \right )}} \\\\\tan{\left (x \right )} = \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \\\\\cot{\left (x \right )} = \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}\)

Now simplify, and combine each term.

\(\displaystyle \\\frac{1}{2} \cos{\left (x \right )} \csc^{2}{\left (x \right )} + \frac{1}{2} \\\\\frac{1}{2} \left(\cos{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)\)

The first step to simplifying is to remember an important trig identity.

\(\displaystyle \sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1\)

If we rewrite it to look like the denominator, it is.

\(\displaystyle \sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1\)

Now we can substitute this in the denominator.

\(\displaystyle \frac{\sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}\)

Now write each term separately.

\(\displaystyle \frac{1}{2} + \frac{\cos^{2}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}\)

Remember the following identities.

\(\displaystyle \\\csc{\left (x \right )} = \frac{1}{\sin{\left (x \right )}} \\\\\sec{\left (x \right )} = \frac{1}{\cos{\left (x \right )}} \\\\\tan{\left (x \right )} = \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \\\\\cot{\left (x \right )} = \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}\)

Now simplify, and combine each term.

\(\displaystyle \\\frac{1}{2} \cos^{2}{\left (x \right )} \csc^{2}{\left (x \right )} + \frac{1}{2} \\\\\frac{1}{2} \left(\cos^{2}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)\)

The first step to simplifying is to remember an important trig identity.

\(\displaystyle \sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1\)

If we rewrite it to look like the denominator, it is.

\(\displaystyle \sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1\)

Now we can substitute this in the denominator.

\(\displaystyle \frac{\sin^{3}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}\)

Now write each term separately.

\(\displaystyle \frac{1}{2} \sin{\left (x \right )} + \frac{\cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}\)

Remember the following identities.

\(\displaystyle \\\csc{\left (x \right )} = \frac{1}{\sin{\left (x \right )}} \\\\\sec{\left (x \right )} = \frac{1}{\cos{\left (x \right )}} \\\\\tan{\left (x \right )} = \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \\\\\cot{\left (x \right )} = \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}\)

Now simplify, and combine each term.

\(\displaystyle \\\frac{1}{2} \sin{\left (x \right )} + \frac{1}{2} \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} \\\\\frac{1}{2} \left(\sin{\left (x \right )} + \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )}\right)\)

The first step to simplifying is to remember an important trig identity.

\(\displaystyle \sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1\)

If we rewrite it to look like the denominator, it is.

\(\displaystyle \sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1\)

Now we can substitute this in the denominator.

\(\displaystyle \frac{\sin^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}\)

Now write each term separately.

\(\displaystyle \frac{1}{2} + \frac{\cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}\)

Remember the following identities.

\(\displaystyle \\\csc{\left (x \right )} = \frac{1}{\sin{\left (x \right )}} \\\\\sec{\left (x \right )} = \frac{1}{\cos{\left (x \right )}} \\\\\tan{\left (x \right )} = \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \\\\\cot{\left (x \right )} = \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}\)

Now simplify, and combine each term.

\(\displaystyle \\\frac{1}{2} \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + \frac{1}{2} \\\\\frac{1}{2} \left(\cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)\)

The first step to simplifying is to remember an important trig identity.

\(\displaystyle \sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1\)

If we rewrite it to look like the denominator, it is.

\(\displaystyle \sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1\)

Now we can substitute this in the denominator.

\(\displaystyle \frac{\sin^{2}{\left (x \right )} + \cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}\)

Now write each term separately.

\(\displaystyle \frac{1}{2} + \frac{\cos^{3}{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}\)

Remember the following identities.

\(\displaystyle \\\csc{\left (x \right )} = \frac{1}{\sin{\left (x \right )}} \\\\\sec{\left (x \right )} = \frac{1}{\cos{\left (x \right )}} \\\\\tan{\left (x \right )} = \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \\\\\cot{\left (x \right )} = \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}\)

Now simplify, and combine each term.

\(\displaystyle \\\frac{1}{2} \cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + \frac{1}{2} \\\\\frac{1}{2} \left(\cos^{3}{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)\)

The first step to simplifying is to remember an important trig identity.

\(\displaystyle \sin^{2}{\left (x \right )} + \cos^{2}{\left (x \right )} = 1\)

If we rewrite it to look like the denominator, it is.

\(\displaystyle \sin^{2}{\left (x \right )} = \cos^{2}{\left (x \right )} - 1\)

Now we can substitute this in the denominator.

\(\displaystyle \frac{\sin^{2}{\left (x \right )} + \cos{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}\)

Now write each term separately.

\(\displaystyle \frac{1}{2} + \frac{\cos{\left (x \right )}}{2 \sin^{2}{\left (x \right )}}\)

Remember the following identities.

\(\displaystyle \\\csc{\left (x \right )} = \frac{1}{\sin{\left (x \right )}} \\\\\sec{\left (x \right )} = \frac{1}{\cos{\left (x \right )}} \\\\\tan{\left (x \right )} = \frac{\sin{\left (x \right )}}{\cos{\left (x \right )}} \\\\\cot{\left (x \right )} = \frac{\cos{\left (x \right )}}{\sin{\left (x \right )}}\)Now simplify, and combine each term.

\(\displaystyle \\\frac{1}{2} \cos{\left (x \right )} \csc^{2}{\left (x \right )} + \frac{1}{2} \\\\\frac{1}{2} \left(\cos{\left (x \right )} \csc^{2}{\left (x \right )} + 1\right)\)