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Cos(a  b)
In trigonometry, cos(a  b) is one of the important trigonometric identities, that finds application in finding the value of the cosine trigonometric function for the difference of angles. The expansion of cos (a  b) helps in representing the cos of a compound angle in terms of trigonometric functions sine and cosine. Let us understand the cos(ab) identity and its proof in detail in the following sections.
1.  What is Cos(a  b) Identity in Trigonometry? 
2.  Cos(a  b) Compound Angle Formula 
3.  Proof of Cos(a  b) Formula 
4.  How to Apply Cos(a  b)? 
5.  FAQs on Cos(a  b) 
What is Cos (a  b) Identity in Trigonometry?
Cos (a  b) is the trigonometric identity for compound angles. We apply the cos (ab) identity formula when the angle for which the value of the cosine function is to be calculated is given in the form of the difference of angles. The angle (ab) represents the compound angle.
cos(a  b) Compound Angle Formula
We refer to cos(a  b) formula as the subtraction formula in trigonometry. The cos(a  b) formula for the compound angle (ab) can be given as,
cos (a  b) = cos a cos b + sin a sin b
Proof of Cos(a  b) Formula
The proof of expansion of cos(ab) formula can be given using the geometrical construction method. Let us see the stepwise derivation of the formula for the cosine trigonometric function of the difference of two angles. In the geometrical proof of cos(ab) formula, we initially assume that 'a' and 'b' are positive acute angles, such that angle a > angle b. This formula, in general, is true for any positive or negative value of a and b.
To prove: cos (a  b) = cos a cos b + sin a sin b
Construction: Draw a line OX in a plane and let us rotate it about O in the anticlockwise direction to a point Z, making an acute ∠XOZ = a, from starting position to its final position. Again, rotate the line, this time in the backward direction, starting from the position OZ till it reaches a point Y, thus making out an acute angle given as, ∠ZOY = b. Therefore, ∠XOY = a  b.
Next, take a point P on OY, and draw PQ and PR perpendiculars to OX and OZ respectively. Draw perpendiculars RS and RT from point R upon OX and PQ respectively.
Now, from the rightangled triangle PQO we get,
cos (a  b) = OQ/OP
= (OS+SQ)/OP
= OS/OP + SQ/OP
= OS/OP + TR/OP
= OS/OR ∙ OR/OP + TR/PR ∙ PR/OP
= cos a cos b + sin ∠TPR sin b
= cos a cos b + sin a sin b, (since we know, ∠TPR = a)
Therefore, cos (a  b) = cos a cos b + sin a sin b.
How to Apply Cos(a  b)?
The expansion of cos(a  b) can be used to find the value of the cosine trigonometric function for angles that can be represented as the difference of standard angles in trigonometry. We can follow the steps given below to learn to apply cos(a  b) identity. Let us evaluate cos(90º  30º) to understand this better.
 Step 1: Compare the cos(a  b) expression with the given expression to identify the angles 'a' and 'b'. Here, a = 90º and b = 30º.
 Step 2: We know, cos (a  b) = cos a cos b + sin a sin b.
⇒ cos(90º  30º) = cos 90ºcos 30º + sin 90ºsin 30º
since, sin 90º = 1, sin 30º = 1/2, cos 90º = 0, cos 30º = √3/2
⇒ cos(90º  30º) = (0)(√3/2) + (1)(1/2) = 0 + 1/2 = 1/2
Also, we know that cos 60º = 1/2. Therefore the result is verified.
Let us have a look a few solved examples to understand cos(ab) formula better.
Examples Using Cos(a  b)

Example 1: Find the exact value of cos 150º using expansion of cos(ab).
Solution:
since, the values of sine and cosine functions can be easily calculated for 30º and 180º, we can write 150º as (180º  30º).
⇒cos(150º) = cos(180º  30º) = cos180ºcos30º + sin 180ºsin30º = (1)(√3/2) + (0)(1/2) = √3/2

Example 2: Apply the cos(a  b) formula to find the expansion of the double angle formula cos 2θ.
Solution:
We can write cos 2θ = cos(θ  (θ))
Applying cos(a  b) = cos a cos b + sin a sin b
cos 2θ = cosθcos(θ) + sinθsin(θ) = cos^{2}θ  sin^{2}θ
∴cos 2θ = cos^{2}θ  sin^{2}θ
FAQs on Cos(a  b)
What is Cos(ab)?
Cos(a  b) is one of the important trigonometric identities, also called the cosine subtraction formula in trigonometry. Cos(ab) can be given as, cos (a  b) = cos a cos b + sin a sin b, where 'a' and 'b' are angles.
What is the Formula of Cos(ab)?
The cos(ab) formula is used to express the cosine compound angle formula in terms of sine and cosine of individual angles. cos(ab) trigonometry formula can be given as, cos (a  b) = cos a cos b + sin a sin b.
What is Expansion of Cos(ab)
The expansion of cos(ab) is given as, cos (a  b) = cos a cos b + sin a sin b. Here, a and b are the measures of angles.
How to Prove Cos(ab) Formula?
The proof of cos(ab) formula can be given using the geometrical construction method. We initially assume that 'a' and 'b' are positive acute angles, such that a > b. Click here to understand the stepwise method to derive cos(ab) formula.
What are the Applications of Cos(ab) Formula?
cos(ab) can be used to find the value of cosine function for angles that can be represented as the difference of standard or simpler angles. Therefore, it makes the deduction easier. It can also be used in finding the expansion of other double and multiple angle formulas.
How to Find the Value of Cos 30º Using Cos (a  b) Identity.
The value of cos 30º using cos(a  b) identity can be calculated by first writing it as cos[(90º  60º)] and then applying cos(a  b) identity.
⇒cos[(90º  60º)] = cos 90ºcos 60º + sin 90ºsin 60º = (0)(1/2) + (1)(√3/2) = 0 + (√3/2) = √3/2.
How to Find Cos(a + b  c) using Cos(a  b)?
We can express cos(a+bc) as cos((a+b)c) and expand using cos(ab), cos(a+b) and sin(a+b) formula as, cos(a+bc) = cos(a+b).cos c + sin(a+b).sin c = cos c.(cos a cos b  sin a sin b) + sin c.(sin a cos b + cos a sin b) = cos a cos b cos c  sin a sin b cos c + sin a cos b sin c + cos a sin b sin c.
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