Pythagorean Identities - Trigonometry

Recognize that is a reworking on , meaning that .

Plug that in to our given equation:

Notice that one of the 's cancel out.

.

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

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

Now we can substitute this in the denominator.

Now write each term separately.

Remember the following identities.

Now simplify, and combine each term.

Recall the Pythagorean Identity:

We can rearrange the terms:

This is exactly what our original equation looks like, so the answer is 1.

To simplify, recognize that is a reworking on , meaning that .

Plug that into our given equation:

Remember that , so .

The expression represents a difference of squares. In this case, the product is (remember that 1 is also a perfect square).

One Pythagoran identity for trigonometric functions is:

Thus, we can say that the most simplified version of the expression is .

There are a couple valid strategies for solving this problem. The simplest is to first factor out from both sides. This leaves us with:

Next, substitute with the known identity to get:

From here, we can eliminate the quadratic by converting:

giving us

Thus,

Write the Pythagorean Identity.

Substitute the value of and solve for .

Since the cosine is in the second quadrant, the correct answer is:

By the Pythagorean identity, the first two terms simplify to 1:

.

Dividing the Pythagorean identity by allows us to simplify the right-hand side.

According to the Pythagorean identity

,

the right hand side of this equation can be rewritten as . This yields the equation

.

Dividing both sides by yields:

.

Dividing both sides by yields:

.

This is precisely the definition of the tangent function; since the domain of consists of all real numbers, the values of which satisfy the original equation also consist of all real numbers. Hence, the correct answer is

.

Step 1: Recall the trigonometric identity that has sine and cosine in it...

The sum is equal to 1.

Using the Pythagorean Identity

,

one can solve for by plugging in for .

Solving for , you get it equal to .

Taking the square root of both sides will get the correct answer of

.

Because ,

therefore:

By definition of cosecant,

There are several ways to work this problem, but all of them use the second Pythagorean trig identity, .

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

Method 1:

Substituting for , we get

, which equals zero.

Method 2:

Substituting for , we get

, 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.

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

Now we can substitute this in the denominator.

Now write each term separately.

Remember the following identities.

Now simplify, and combine each term.

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

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

Now we can substitute this in the denominator.

Now write each term separately.

Remember the following identities.

Now simplify, and combine each term.

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

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

Now we can substitute this in the denominator.

Now write each term separately.

Remember the following identities.

Now simplify, and combine each term.

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

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

Now we can substitute this in the denominator.

Now write each term separately.

Remember the following identities.

Now simplify, and combine each term.

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

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

Now we can substitute this in the denominator.

Now write each term separately.

Remember the following identities.

Now simplify, and combine each term.

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

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

Now we can substitute this in the denominator.

Now write each term separately.

Remember the following identities.

Now simplify, and combine each term.

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

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

Now we can substitute this in the denominator.

Now write each term separately.

Remember the following identities.

Now simplify, and combine each term.