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# Trigonometric Identities

## Reciprocal Identities

The trigonometric ratios cosecant, secant, and cotangent are reciprocals of sine, cosine, and tangent, respectively.

### Proof

Consider a right triangle with the three sides labeled with respect to an acute angle

Next, the sine, cosine, tangent, cosecant, secant, and cotangent ratios are written.
The reciprocal of the sine ratio will now be calculated.
Solve for
It has been found that which is the reciprocal of is equal to By the definition, the cosecant of is also the ratio of the lengths of the hypotenuse and the opposite side to Therefore, by the Transitive Property of Equality, is equal to
By following a similar procedure, the other two identities for secant and cotangent can be proven.

## Tangent and Cotangent Identities

The tangent of an angle can be expressed as the ratio of the sine of to the cosine of

Similarly, the cotangent of can be expressed as the ratio of the cosine of to the sine of

### Proof

Tangent Identity

Two proofs will be written for this identity, one using a right triangle and the other using a unit circle.

### Right Triangle

In a right triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side

At the same time, the sine and cosine of can be written as follows.
By manipulating the right-hand side of the equation the tangent can be expressed as the sine over the cosine of The proof of the identity is complete.

### Unit Circle

Consider a unit circle and an angle in standard position.

It is known that the point of intersection of the terminal side of the angle and the unit circle has coordinates

Draw a right triangle using the origin and as two of its vertices. The length of the hypotenuse is and the lengths of the legs are and

As shown previously, the tangent of a right triangle is defined as the ratio of the length of the opposite side — in this case, — to the length of the adjacent side, which here is

### Proof

Cotangent Identity

Two more proofs will be written for this identity, one of them using just a right triangle and the other using a unit circle.

### Right Triangle

In a right triangle, the cotangent of an angle is defined as the ratio of the length of the adjacent side to the length of the opposite side

Additionally, the sine and cosine of can be written as follows.
By manipulating the right-hand side of the equation the cotangent can be expressed as the cosine over the sine of This proof is complete.

### Unit Circle

Using a unit circle, it has been already proven that the tangent of an angle is the ratio of the sine to the cosine of the angle
By manipulating the above equation, it can be shown that the cotangent of is the ratio of the cosine of to the sine of

This proof is complete.

## Pythagorean Identities

For any angle the following trigonometric identities hold true.

### Proof

For Acute Angles
Consider a right triangle with a hypotenuse of
By recalling the sine and cosine ratios, the lengths of the opposite and adjacent sides to can be expressed in terms of the angle.
Definition Substitute Simplify

It can be seen that if the hypotenuse of a right triangle is the sine of an acute angle is equal to the length of its opposite side. Similarly, the cosine of the angle is equal to the length of its adjacent side.

By the Pythagorean Theorem, the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. Therefore, for the above triangle, the sum of the squares of and is equal to the square of

Since represents a side length, it is not Therefore, by diving both sides of the above equation by the second identity can be obtained.
Simplify

The second identity was obtained.
Since represents a side length, it is not Therefore, by dividing both sides of by the third identity can be proven.
Simplify

Finally, the third identity was obtained.

### Proof

For Any Angle

The first identity can be shown using the unit circle and the Pythagorean Theorem. Consider a point on the unit circle in the first quadrant, corresponding to the angle A right triangle can be constructed with

By the Pythagorean Theorem, the sum of the squares of and equals
In fact, this is true not only for points in the first quadrant, but for every point on the unit circle. Recall that, for points on the unit circle corresponding to angle it is known that and that By substituting these expressions into the equation, the first identity can be obtained.
Dividing both sides by either or leads to two variations of the Pythagorean Identity.

## Negative Angle Identities

The function has odd symmetry, and has even symmetry, which can be seen from their graphs. As a result, the corresponding identities hold true.

### Proof

and

The identities will be proven using a unit circle. Consider an arbitrary on a unit circle. Let be the point that the angle forms on the circle.

Recall that the values on the axis are represented by cosine and the value on the axis are represented by sine. Therefore, the coordinates of are

Next, the point can be reflected over the axis.
Since is the same distance from the axis as its coordinate is also Note that a reflection is a congruent transformation, so and are equidistant from the axis too. However, is below the axis, so its coordinate is a negative

Additionally, after the reflection, the angle between the origin, and the axis is This means that the lengths of the newly created horizontal and vertical segments are and Therefore, the coordinates of can also be written as

This way there are two pairs of coordinates of the same point This allows to conclude the following.

### Proof

By expressing using sine and cosine, this identity can be shown.

## Cofunction Identities

For any angle the following trigonometric identities hold true.

### Proof

For Acute Angles

Consider a right triangle. The measure of its right angle is or radians. Let be the radian measure of one of the acute angles. Since the sum of two acute angles in a right triangle is the measure of the third acute angle must be

Let also and represent the side lengths of the triangle. In this case, cosine of can be expressed as the ratio of the lengths of the angle's adjacent side and the hypotenuse.
At the same time, sine of the opposite angle can be expressed as the ratio of the lengths of the angle's opposite side and the hypotenuse.
Since the right-hand sides of the equations are equal, by the Transitive Property of Equality, the left-hand sides are also equal.

This identity is true for all angles, not just those that make it possible to construct a right triangle. Using similar reasoning, the corresponding identities for cosine and tangent can be proven.

### Proof

For Any Angle
The given identities can be proven using the Angle Sum and Difference Identities for any angle measure. Consider the identity for the sine of the difference between two angles.
This identity will be applied to the left-hand side of the first identity.

Therefore, the Cofunction Identity for sine was obtained.

## Angle Sum and Difference Identities

To evaluate trigonometric functions of the sum of two angles, the following identities can be applied.

There are also similar identities for the difference of two angles.

These identities are useful when finding the exact value of the sine, cosine, or tangent of a given angle.

### Proof

Let be a right triangle with hypotenuse and an acute angle with measure

By definition, the sine of an angle is the ratio between the lengths of the opposite side and the hypotenuse.
The idea now is to rewrite in terms of and To do it, draw a ray so that is divided into two angles with measures and Let be a point on this ray such that and are right triangles.
Consider By calculating the sine and cosine of the legs of this triangle can be rewritten.
Now consider Knowing that the sine of can be used to write in terms of and
Solve for
Let be the point of intersection between and Notice that by the Vertical Angles Theorem.

By the Third Angle Theorem, it is known that Therefore,

Since the purpose is to rewrite plot a point on such that This way a rectangle is formed. The opposite sides of a rectangle have the same length, so and are equal. Also, makes a right triangle.

Consequently, and can be written in terms of and using the cosine ratio.
Finally, by the Segment Addition Postulate, is equal to the sum of and All these lengths have been rewritten in terms of the sine and cosine of and
This concludes the proof of the first identity. The other identities can be proven using similar reasoning.

### Extra

Calculating

Consider the following process for calculating the exact value of

1. To be able to use the angle sum identities, the angle needs to be rewritten as the sum of two angles for which the sine and cosine are known. For example, can be rewritten as
2. Use the first formula for the angle sum.
3. Based on the trigonometric ratios of common angles, it is known that and
Following these three steps, the value of can be found.

Simplify
Notice that could also be rewritten as because and are known values.

## Double-Angle Identities

The double-angle identities materialize when two angles with the same measure are substituted into the angle sum identities.

These identities simplify calculations when evaluating trigonometric functions of twice an angle measure.

### Proof

Double-Angle Identities
Start by writing the Angle Sum Identity for sine and cosine.
Let and With this, becomes Then, these two formulas can be rewritten in terms of

### Sine Identity

Approaching the first equation, the Commutative Property of Multiplication can be applied to the second term of its right-hand side. Then, by adding the terms on the right-hand side of this equation, the formula for is obtained.

### Cosine Identities

Approaching the second equation, the Product of Powers Property can be used to rewrite its right-hand side. By doing this, the first identity for the cosine of the double of an angle is obtained.

Now, recall that, by the Pythagorean Identity, the sine square plus the cosine square of the same angle equals From this identity, two different equations can be set.
Next, substitute Equation (I) into the first identity for the cosine.
Substitute for and simplify
That way, the second identity for the cosine has been obtained. To obtain the third cosine identity, substitute Equation (II) into the first identity for the cosine.
Substitute for and simplify

### Tangent Identity

To prove the tangent identity, start by rewriting in terms of sine and cosine.
Next, substitute the first sine identity in the numerator and the first cosine identity in the denominator.
Then, divide the numerator and denominator by
Finally, simplifying the right-hand side the tangent identity will be obtained.
Simplify right-hand side

### Extra

Calculating

To calculate the exact value of these steps can be followed.

1. To be able to use the double-angle identities, the angle needs to be rewritten as multiplied by another angle. Therefore, rewrite as
2. Use the second formula for the cosine of twice an angle.
3. Based on the trigonometric ratios of common angles, it is known that
Following these three steps, the value of can be found.

Simplify

## Half-Angle Identities

The half-angle identities are special cases of angle difference identities. To evaluate trigonometric functions of half an angle, the following identities can be applied.

The sign of each formula is determined by the quadrant where the angle lies.

These identities are useful when finding the exact value of the sine, cosine, or tangent at a given angle.

### Proof

Half-Angle Identities
First, write two of the Double-Angle Identities for cosine.

### Sine Identity

Start by solving the first identity written above for
Solve for
Next, substitute for to obtain the half-angle identity for the sine.

### Cosine Identity

Start by solving the second identity written at the beginning for
Solve for
Next, substitute for to obtain the half-angle identity for the cosine.

### Tangent Identity

To derive the tangent identity, start by recalling the definition of the tangent ratio.
Next, substitute for
Finally, substitute the half-identities for the sine and cosine into the equation above and simplify the right-hand side.
Simplify right-hand side

### Extra

Calculating

Consider the calculation of the exact value of

1. To be able to use the half-angle identities, the angle needs to be rewritten as a certain angle divided by Therefore, rewrite as
2. Based on the trigonometric ratios of common angles, it is known that
3. According to the diagram of the quadrants, an angle that measures is in the first quadrant. Therefore, the cosine ratio is positive.
With these three steps and the second identity in mind, the value of can be found.

Simplify