Sum and Difference Identities Calculator

Trigonometric sum and difference formulas allow you to evaluate functions of angles that can be expressed as the sum or difference of two known, standard angles (such as 30°, 45°, and 60°). For sine, the fundamental formulas are sin(α ± β) = sin(α)cos(β) ± cos(α)sin(β). For cosine, the rules change slightly regarding signs: cos(α ± β) = cos(α)cos(β) ∓ sin(α)sin(β). Tangent identities build upon these by dividing the sine identity result by the cosine identity result, yielding tan(α ± β) = (tan(α) ± tan(β)) / (1 ∓ tan(α)tan(β)). Reciprocal functions such as cosecant, secant, and cotangent are derived directly by taking the reciprocal of these primary sine, cosine, and tangent values.

To find the exact value of cos(15°) using the difference identity, we can express 15° as the difference between two familiar benchmark angles: 45° and 30° (since 45° - 30° = 15°). Here, let α = 45° and β = 30°.

1. Apply the cosine difference formula: cos(α - β) = cos(α)cos(β) + sin(α)sin(β).

2. Substitute the known values: cos(45° - 30°) = cos(45°)cos(30°) + sin(45°)sin(30°).

3. Insert the exact trigonometric ratios: cos(45°) = √2 / 2, cos(30°) = √3 / 2, sin(45°) = √2 / 2, and sin(30°) = 1 / 2.

4. Multiply and add the terms: ( (√2 / 2) * (√3 / 2) ) + ( (√2 / 2) * (1 / 2) ) = (√6 / 4) + (√2 / 4).

5. Combine over a common denominator to get the final exact expression: (√6 + √2) / 4.

When working with trigonometric identities, always double-check your angle unit settings—ensure your calculator or manual notes are properly configured for degrees or radians. When simplifying expressions algebraically, watch the signs closely; remember that the cosine difference formula uses a plus sign in the middle, while the cosine sum formula uses a minus sign. Finally, leverage exact radical forms rather than rounded decimals whenever possible to preserve mathematical precision in higher-level physics and calculus applications.