Trigonometric identities are equalities using trigonometric functions that hold true for any value of the variables involved, hence defining both sides of the equality. These are equations that relate to various trigonometric functions and are true for every variable value in the domain. The formulae sin(A+B), cos(A-B), and tan(A+B) are some of the sum and difference identities.

**Tangent Trigonometric Ratio**

The ratio of any two right triangle sides is called a trigonometric ratio. The tangent ratio is defined as the ratio of the length of the opposite side of an angle divided by the length of the adjacent side.

If θ is the angle created by the base and hypotenuse of a right-angled triangle then,

tan θ = Perpendicular/Base = sin θ/ cos θ

Here, perpendicular is the side opposite to the angle and base is the side adjacent to it.

**Tangent Addition Formula**

In trigonometry, the tangent addition formula is referred to as the tan(A + B) formula for the compound angle (A+B). It is used when the angle for which the tangent function value is to be determined is supplied as the sum of any two angles. It may alternatively be written as tan(A + B) = sin (A + B)/cos (A + B) since the tangent function is a ratio of the sine and cosine functions.

tan(A + B) = (tan A + tan B)/(1 – tan A tan B)

**Derivation**

The formula for tangent addition is derived by using the formulas for expansion of sum angle for sine and cosine ratios.

Now, we know that,

tan (A + B) = sin (A + B)/cos (A + B) …… (1)

Substitute sin (A + B) = sin A cos B + cos A sin B and cos (A + B) = cos A cos B – sin A sin B in the equation (1).

tan (A + B) = (sin A cos B + cos A sin B)/(cos A cos B – sin A sin B)

Dividing the numerator and denominator by cos A cos B, we get

tan (A + B) = [(sin A cos B + cos A sin B)/(cos A cos B)]/[(cos A cos B – sin A sin B)/(cos A cos B)]

tan (A + B) = [(sin A cos B)/(cos A cos B) + (cos A sin B)/(cos A cos B)]/[(cos A cos B)/(cos A cos B) – (sin A sin B)/(cos A cos B)]

tan (A + B) = (tan A + tan B)/(1 – tan A tan B)

This derives the formula for tangent addition of any two angles, A and B.

**Sample Problems**

**Problem 1. If tan A = 1/2 and tan B = 1/3, find the value of tan (A+B) using the formula.**

**Solution:**

We have, tan A = 1/2 and tan B = 1/3.

Using the formula we get,

tan (A + B) = (tan A + tan B)/(1 – tan A tan B)

= (1/2 + 1/3)/(1 – (1/2)(1/3))

= (5/6)/(1 – 1/6)

= (5/6)/(5/6)

= 1

**Problem 2. If tan A = 2/3 and tan B = 4/7, find the value of tan (A+B) using the formula.**

**Solution:**

We have, tan A = 2/3 and tan B = 4/7.

Using the formula we get,

tan (A + B) = (tan A + tan B)/(1 – tan A tan B)

= (2/3 + 4/7)/(1 – (2/3)(4/7))

= (16/21)/(1 – 8/21)

= (16/21)/(13/21)

= 16/13

**Problem 3. If tan (A+B) = 15/11 and tan A = 2/11, find the value of tan B using the formula.**

**Solution:**

We have, tan (A+B) = 15/11 and tan A = 2/11.

Let tan B = x.

Using the formula we get,

tan (A + B) = (tan A + tan B)/(1 – tan A tan B)

=> 15/11 = (2/11 + x)/(1 – (2/11)(x))

=> 15/11 = ((2 + 11x)/11)/((11 – 2x)/11)

=> 15/11 = (11x + 2)/(11 – 2x)

=> 165 – 30x = 121x + 22

=> 151x = 143

=> x = 143/151

=> tan B = 143/151

**Problem 4. If tan B = 6/13 and tan (A+B) = 9/13, find the value of tan A using the formula.**

**Solution:**

We have, tan (A+B) = 9/13 and tan B = 6/13.

Let tan A = x.

Using the formula we get,

tan (A + B) = (tan A + tan B)/(1 – tan A tan B)

=> 9/13 = (x + 6/13)/(1 – (x)(6/13))

=> 9/13 = ((13x + 6)/13)/((13 – 6x)/13)

=> 9/13 = (13x + 6)/(13 – 6x)

=> 117 – 54x = 169x + 78

=> 223x = 39

=> x = 39/223

=> tan A = 39/223

**Problem 5. If sin A = 4/5 and cos B = 5/13, find the value of tan (A+B) using the formula.**

**Solution:**

We have, sin A = 4/5.

It means, cos A = 3/5. So, tan A = 4/3.

Also, cos B = 5/13.

It means, sin B = 12/13. So, tan B = 12/5.

Using the formula we get,

tan (A + B) = (tan A + tan B)/(1 – tan A tan B)

= (4/3 + 12/5)/(1 – (4/3)(12/5))

= (56/15)/(1 – 48/15)

= (56/15)/(-33/15)

= -56/33

## Trigonometric Addition Formulas

Angle addition formulas express trigonometric functions of sums of angles

formulas of angle addition in trigonometry are given by

The first four of these are known as the prosthaphaeresis formulas, or sometimes as Simpson’s formulas.

The sine and cosine angle addition identities can be compactly summarized by the matrix equation

These formulas can be simply derived using complex exponentials and the Euler formula as follows.

(8) | |||

(9) | |||

(10) | |||

Multiple-angle formulas are given by

The angle addition formulas can also be derived purely algebraically without the use of complex numbers. Consider the small right triangle in the figure above, which gives

## Tangent Sum and Difference Formulas

## Tangent Identities

Formulas for the tangent function can be derived from similar formulas involving the sine and cosine. The sum identity for tangent is derived as follows:

**difference identity for tangent**, use the fact that tan(−β) = −tanβ

**Example 1:** Find the exact value of tan 75°.

Because 75° = 45° + 30°

**Example 2:**Verify that tan (180° −

*x*) = −tan

*x*.

**Example 3:**Verify that tan (180° +

*x*) = tan

*x*.

The preceding three examples verify three formulas known as the **reduction identities for tangent**. These reduction formulas are useful in rewriting tangents of angles that are larger than 90° as functions of acute angles.

The **double‐angle identity for tangent** is obtained by using the sum identity for tangent.

**half‐angle identity for tangent**can be written in three different forms.

In the first form, the sign is determined by the quadrant in which the angle α/2 is located.

**Example 5:** Verify the identity

**Example 6**: Verify the identity tan (α/2) = (1 − cos α)/sin α.

**Example 7:** Verify the identity tan (α − 2) = sin π/(1 + cos α).

Begin with the identity in Example 6.

**Example 8:** Use a half‐angle identity for the tangent to find the exact value for tan 15°.

What follows are two alternative solutions.

## What Are Tangent Formulas?

The tangent formulas talk about the tangent (tan) function. Let us consider a right-angled triangle with one of its acute angles to be x. Then the tangent formula is, tan x = (opposite side) / (adjacent side), where “opposite side” is the side opposite to the angle x, and “adjacent side” is the side that is adjacent to the angle x. Apart from this general formula, there are so many other formulas in trigonometry that will define a tangent function which you can see in the following image.

### Tangent Formulas Using Reciprocal Identity

We know that the tangent function (tan) and the cotangent function (cot) are reciprocals of each other. i.e., if tan x = a / b, then cot x = b / a. Thus, tangent formula using one of the reciprocal identities is,

**tan x = 1 / (cot x)**

### Tangent Formula Using Sin and Cos

We know that sin x = (opposite) / (hypotenuse), cos x = (adjacent) / (hypotenuse), and tan x = (opposite) / (adjacent). Now we will divide sin x by cos x.

(sin x) / (cos x) = [ (opposite) / (hypotenuse) ] / [ (adjacent) / (hypotenuse) ] = (opposite) / (adjacent) = tan x

Thus, the tangent formula in terms of sine and cosine is,

**tan x = (sin x) / (cos x)**

### Tangent Formulas Using Pythagorean Identity

One of the Pythagorean identities talks about the relationship between sec and tan. It says, sec^{2}x – tan^{2}x = 1, for any x. We can solve this for tan x. Let us see how.

sec^{2}x – tan^{2}x = 1

Subtracting sec^{2}x from both sides,

-tan^{2}x = 1 – sec^{2}x

Multiplying both sides by -1,

tan^{2}x = sec^{2}x – 1

Taking square root on both sides,

**tan x = ± √( sec ^{2}x – 1)**

### Tangent Formula Using Cofunction Identities

The cofunction identities define the relation between the cofunctions which are sin, cos; sec, csc; and tan, cot. Using one of the cofunction identities,

- tan x = cot (90
^{o}– x) (OR) - tan x = cot (π/2 – x)

### Tangent Formulas Using Sum/Difference Formulas

We have sum/difference formulas for every trigonometric function that deal with the sum of angles (A + B) and the difference of angles (A – B). The sum/difference formulas of tangent function are,

- tan (A + B) = (tan A + tan B) / (1 – tan A tan B)
- tan (A – B) = (tan A – tan B) / (1 + tan A tan B)

### Tangent Formula of Double Angle

We have double angle formulas in trigonometry which deal with 2 times the angle. The double angle formula of tan is

tan 2x = (2 tan x) / (1 – tan^{2}x)

### Tangent Formula of Triple Angle

We have triple angle formulas for all trigonometric functions. Among them, the triple angle formula of the tangent function is,

tan 3x = (3 tan x – tan^{3}x) / (1 – 3 tan^{2}x)

### Tangent Formula of Half Angle

We have half-angle formulas in trigonometry that deal with half of the angles (x/2). The half-angle formulas of tangent function are,

- tan (x/2) =± √[ (1 – cos x) / (1 + cos x) ]
- tan (x/2) = (1 – cos x) / ( sin x)

## Examples Using Tangent Formulas

**Example 1: **If sec x = 5/3 and x is in the first quadrant, find the value of tan x.

**Solution:**

Using one of the tangent formulas,

tan x = ± √(sec^{2}x – 1)

Since x is in the first quadrant, cos x is positive. Thus,

tan x = ± √(sec^{2}x – 1)

Substitute sec x = 5/3 here,

tan x = √((5/3)^{2} – 1)

= √((25/9) – 1)

=√ (16/9)

= 4/3

**Answer: **tan x = 4/3.

**Example 2: **If cot (90 – A) = 3/2, then find the value of tan A.

**Solution:**

Using one of the tangent formulas,

tan A = cot (90 – A)

It is given that cot (90 – A) = 3/2. Hence,

tan A = 3/2

**Answer: **tan A = 3/2.

**Example 3: **If tan A = 1/2 and tan B = 1/3, find tan (A + B).

**Solution:**

Using the tangent formula of addition,

tan (A + B) = (tan A + tan B) / (1 – tan A tan B)

= (1/2 + 1/3) / ( 1 – (1/2) · (1/3) )

= (5/6) / (1 – (1/6))

= (5/6) / (5/6)

= 1

**Answer: **tan (A + B) = 1.

## FAQs on Tangent Formulas

### What Are Tangent Formulas?

The tangent formulas are related to the tangent function. The important tangent formulas are as follows:

- tan x = (opposite side) / (adjacent side)
- tan x = 1 / (cot x)
- tan x = (sin x) / (cos x)
- tan x = ± √( sec
^{2}x – 1)

### How To Derive Tangent Formula of Sum?

The tangent formula of sum/addition is, tan (A + B) = (tan A + tan B) / (1 – tan A tan B). Let us derive this starting with the left side part.

tan (A + B) = sin (A + B) / cos (A + B)

= [ sin A cos B + cos A sin B ] / [cos A cos B – sin A sin B]

Dividing each term in both numerator and denominator by cos A cos B,

tan (A + B) = [ (sin A / cos A) + (sin B / cos B) ] / [ 1 – (sin A / cos A) (sin B / cos B) ]

= (tan A + tan B) / (1 – tan A tan B)

### What Are the Applications of Tangent Formulas?

As we have learned on this page, we have multiple tangent formulas and we can choose one of them to prove a trigonometric identity (or) find the value of the tangent function with the available information. We also use tangent formulas in Calculus.

### How To Derive the Double Angle Tangent Formula?

Using the sum formula of tangent function, we have, tan (A + B) = (tan A + tan B) / (1 – tan A tan B). Substituting A = B on both sides here, we get,

tan 2x = (tan x + tan x) / (1 – tan x · tan x) = (2 tan x) / (1 – tan^{2}x).

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