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More Trigonometric Identities

In these lessons, solutions, and examples we will learn**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:**

**Example:**

**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 - 16sin^{2}x cos^{2}x

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.

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.

**Solution:**

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

**Solution:**

Example:

1. Find sin(105°) exactly

2. Find cos(105°) exactly

3. Find tan(105°) exactly

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)

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 - 16sin

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