Sum And Difference Identities

Related Pages
Lessons On Trigonometry
Inverse trigonometry
Trigonometric Functions

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

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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)

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How to use the Sum and Difference Identities to Prove Other Identities

Example:
Prove sin(α + β) - sin(α - β) = 2cosαsinβ

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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)

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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)

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Using the Sum and Difference Identities for Sine, Cosine and Tangent

Example 3:
Simplify 1 - 16sin²x cos²x

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