# Sum and Difference Identities

Related Topics: More Trigonometric Identities

In these lessons, solutions, and examples we will learn
• the sum identities and difference identities for sine, cosine and tangent.
• how to use the sum identities and difference identities to simplify trigonometric expressions.
• how to use the sum identities and difference identities to prove other trigonometric identities.

What are the Sum and Difference Identities?

The following shows the Sum and Difference Identities for sin, cos and tan. Scroll down the page for more examples and solutions on how to use the identities.

Example:

Solution:

Given that cos(α + β) = cos α cos β – sin α sin β, then

Example:

Solution:

How to use the sum and difference identities for sin, cos, and tan?
Example:
1. Find sin(105°) exactly
2. Find cos(105°) exactly
3. Find tan(105°) exactly How to use Sum and Difference Identities to find exact trig values?
Example:
1. Find $$\cos \left( {\frac{{3\pi }}{4},\frac{\pi }{3}} \right)$$ exactly

2. Find cos(42°)cos(18°) - sin(42°)sin(18°) exactly

3. Find $$\frac{{\tan 80^\circ - \tan 35^\circ }}{{1 + \tan 80^\circ \tan 35^\circ }}$$ exactly

4. Find cos(u + v) exactly if sin(u) = 3/5 and sin(v) = 12/13 where u and v are acute angles (quadrant I) How to use the Sum and Difference Identities to Prove Other Identities
Examples:
Prove sin(α + β) - sin(α - β) = 2cosαsinβ
Using the Sum and Difference Identities for Sine, Cosine and Tangent
Example 1:
If sin x = 12/13 and x is in the first quadrant, find tan(2x)

Using the Sum and Difference Identities for Sine, Cosine and Tangent
Example 2:
If tan x = 5/3 and x is in the first quadrant, find sin(2x) Using the Sum and Difference Identities for Sine, Cosine and Tangent
Example 3:
Simplify 1 - 16sin2x cos2x

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