We can multiply the bottom with its conjugate and obtain:

\(\displaystyle \frac{1}{1 + cos \ x} * \frac{1- cos \ x}{1 - cos \ x} = \frac{1 - cos x}{1 - cos^2 x}\)

Then we can use the pythagorean identity for the cosines and sines:

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

Finally, we can split the fractions up and translate them into the trigonometric identity:

\(\displaystyle = \frac{1}{sin^2x} - \frac{cos \ x}{sin^2x} = (\frac{1}{sin \ x})^2 - \frac{cos \ x}{sin \ x}*\frac{1}{sin \ x}\)

\(\displaystyle \mathbf{= csc^2 x - cot \ x \ csc \ x}\)

Alternatively, you could take this and other answer choices and work the opposite way by translating all of the trigonometric ratios into sines and cosines, using the identities.

Another potential option is to try certain angle values for the original identity and selecting the choice that most closely matches the answer choices when the same angle is substituted (although one must beware of where the function(s) are undefined).

The power-reducing formulas state that:

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

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

\(\displaystyle tan^2(x)=\frac{1-cos(2x)}{1+cos(2x)}\)

To find the mean of a certain set of numbers we need to add all enteries together and then divide by how many numbers you have.

In our case we do:

\(\displaystyle \frac{87+92+77+90}{4}\)

\(\displaystyle \frac{346}{4}\)

\(\displaystyle 86.5\)

For this question we need to set up an equation to find the mean, set it equal to our desired mean of 92 and solve for our missing test value.

\(\displaystyle \frac{93+88+87+95+x}{5}=92\)

From here we perform algebraic procedures to simplify the equation

\(\displaystyle \frac{363+x}{5}=92\)

\(\displaystyle 363+x=460\)

\(\displaystyle x=97\)

To solve this problem we use the equation to find the mean. We add all entries and then divide by the number of entries we have. It is as follows:

\(\displaystyle \frac{10+9+5}{3}=\frac{24}{3}\)

\(\displaystyle \frac{24}{3}=8\)

The mean is the same as the average. To find the mean, use the following formula:

\(\displaystyle \text{Mean}=\frac{\text{Sum of all values}}{\text{Number of Values}}\)

\(\displaystyle \text{Mean}=\frac{100+90+67+89+78}{5}=84.8\)

First, write out the individual values found within this stem and leaf plot to figure out how many values there are.

\(\displaystyle \text{Values}: 20, 21, 24, 24, 30, 31, 31, 32, 32, 35, 37, 41, 45\)

Now, finding the mean is the same as finding the average. Recall how to find the average:

\(\displaystyle \text{Mean/Average}=\frac{\text{Sum of all values}}{\text{Number of values}}\)

Thus, the mean number of baseball cards owned is found by the following:

\(\displaystyle \text{Mean}=\frac{20+21+24+24+30+31+31+32+32+35+37+41+45}{13}=31\)

Recall how to find the average of a set of numbers:

\(\displaystyle \text{Mean/Average}=\frac{\text{Sum of all values}}{\text{Number of values}}\)

Now, let \(\displaystyle x\) be the score Michael needs on his last test in order to have a test average of \(\displaystyle 80\).

\(\displaystyle 80=\frac{65+90+78+67+x}{5}\)

Now, solve for \(\displaystyle x\).

\(\displaystyle 300+x=400\)

\(\displaystyle x=100\)

First, figure out the temperatures.

May: \(\displaystyle 69\)

June:\(\displaystyle 78\)

July: \(\displaystyle 88\)

August: \(\displaystyle 86\)

September:\(\displaystyle 79\)

Now, recall how to find the mean of a set of numbers:

\(\displaystyle \text{Mean/Average}=\frac{\text{Sum of all values}}{\text{Number of values}}\)

\(\displaystyle \text{Mean}=\frac{69+78+88+86+79}{5}=80\)

Recall how to find the mean:

\(\displaystyle \text{Mean/Average}=\frac{\text{Sum of all values}}{\text{Number of values}}\)

\(\displaystyle \frac{x+(x-1)+(x+7)+(x+10)}{4}=9\)

\(\displaystyle 4x+16=36\)

\(\displaystyle 4x=20\)

\(\displaystyle x=5\)