Cos (a + b) is one of the important trigonometric identities, cos (a + b) is also called the cosine addition formula in trigonometry. Cos(a+b) is given as, cos (a + b) = cos a cos b – sin a sin b. In this article, we will learn about, cos(a + b), Proof of this Identity, How to Apply cos(a + b) Formula, and Others in detail.

Formula for Cos (a + b) is,

Cos(a + b) = cos a cos b – sin a sin b

Cos (a+ b) Formula

Trigonometry Identities

Trigonometric identities are equations involving trigonometric functions that are true for all possible values of the variables within their domains.

These identities are fundamental in trigonometry and are used to simplify expressions, solve equations, and establish relationships between different trigonometric functions.

What is Cos(a + b)?

The trigonometric identity for the cosine of a sum of two angles is expressed as cos(a + b) = cos(a)cos(b) – sin(a)sin(b). This identity is used to find the cosine of a compound angle, where the angle is given as the sum of two separate angles. In this context, (a + b) represents the compound angle.

Cos(a + b) Formula

Cos(A + B) is a trigonometric identity for compound angles. We use this identity when the angle for which we want to calculate the cosine function is given as the difference of two angles, such as (90° + 30°) or (45° + 15°). Angle (A + B) represents the compound angle.

Cos(a + b) Compound Angle Formula

Formula for cos(A + B) is given by,

cos(A + B) = cos A.cos B – sin A.sin B

This formula can express the cosine of a compound angle in terms of the sine and cosine functions of the individual angles.

Some Similar Formulas to Cos(a+b)

Some similar formulas to Cos (a+ b) Formula are:

• sin(a+b) = sin a cos b − cos a sin b
• cos (a – b) = cos a cos b + sin a sin b

Cos(a + b) Formula Proof

Proof of cos(A + B) formula can be done using various methods, such as Geometrical Construction Method, Using Complex Numbers, etc. Proof of cos(A + B) using complex numbers is discussed below,

Cos(A + B) formula can be derived using the complex numbers as,

e^ix = cos x + i.sin x

Let us assume x = (A + B)

e^i(A+B) = cos (A+B) + i.sin (A+B)

Now, applying exponent rule in e^i(A+B)

e^i(A+B) = e^i(A). e^i(B)

cos (A+B) + i.sin (A+B)

= {cos A + i.sin A}.{cos (B) + i.sin (B)}

= cos A.cos B + i.cos A.sin B + isin A.cos B + i²sinA.sinB

[ Here i = √-1 and i² = -1 ]

= cos A.cos B – sinA.sinB + i(cos A.sin B +sin A.cos B)

Comparing Real and Imaginary Parts,

• cos (A + B) = cos A.cos B – sin A.sin B
• sin (A + B) = sin A.cos B + cos A.sin B

Thus, cos (A+B) formula is derived.

How to Apply Cos(a + b) Formula?

Cos(A + B) formula can be used to find the value of cosine function for angles that can be expressed as the sum of standard or simpler angles. For example, we can use this formula to find cos(75°) which is not directly available in the trigonometric table.

To apply the cos(A + B) formula, we can follow simple three steps given below:

Step 1: First we need to figure out the angles A and B in the given expression, so that A + B is equal to the required angle say theta.

Step 2: Substitute the values of sin and cos of A and B from trigonometric table or using other identities.

Step 3: Simplify the expression to get the final answer.