Advanced Trigonometric Identities

Trigonometric identities are powerful tools for simplifying expressions, solving equations, and proving mathematical statements.

The following tables give some Trigonometric Identities. Scroll down the page for examples and solutions on how to use the Trig Identities.

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1. Sum Identities
   These identities allow you to express trigonometric functions of the sum of two angles in terms of trigonometric functions of the individual angles:
   \( sin(A + B) = sinAcosB + cosAsinB \)
   \( cos(A + B) = cosAcosB − sinAsinB \)
   \( tan(A + B) = \frac{tanA + tanB}{1- tanAtanB} \)
   

2. Difference Identities
   These identities allow you to express trigonometric functions of the difference of two angles in terms of trigonometric functions of the individual angles:
   \( sin(A - B) = sinAcosB - cosAsinB \)
   \( cos(A - B) = cosAcosB + sinAsinB \)
   \( tan(A - B) = \frac{tanA - tanB}{1 + tanAtanB} \)
   

3. Double-Angle Identities
   These are special cases of the sum identities where the two angles are equal (A = B = θ):
   \( sin(2θ) = 2sinθcosθ \)
   \( cos(2θ) = cos^2θ−sin^2θ \)
   \( cos(2θ) = 1 − 2sin^2θ \)
   \( cos(2θ) = 2cos^2θ − 1 \)
   \( tan(2θ) = \frac{2tanθ}{1 + tan^2θ} \)
   

4. Half-Angle Identities
   These identities express trigonometric functions of an angle that is half of another angle \( \frac{θ}{2} \) in terms of trigonometric functions of the full angle \( θ \):
   \( sin(\frac{\theta}{2}) = \pm \sqrt{\frac{1 - \cos \theta}{2}} \)
   \( cos\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}} \)
   \( tan\left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}} = \frac{\sin \theta}{1 + \cos \theta} = \frac{1 - \cos \theta}{\sin \theta} \)

5. Product-to-Sum Identities
   These identities allow you to rewrite the product of two trigonometric functions as a sum or difference:
   \( sin A \cos B = \frac{1}{2} [\sin(A + B) + \sin(A - B)] \)
   \( cos A \sin B = \frac{1}{2} [\sin(A + B) - \sin(A - B)] \)
   \( cos A \cos B = \frac{1}{2} [\cos(A + B) + \cos(A - B)] \)
   \( sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)] \)
   

6. Sum-to-Product Identities
   These identities do the reverse of the product-to-sum identities, allowing you to rewrite the sum or difference of two trigonometric functions as a product:
   \( sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \)
   \( sin A - \sin B = 2 \cos\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \)
   \( cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \)
   \( cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right) \)
   

Using a Cofunction Identity
Cofunction identities and how to determine cofunctions given a function value.

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Cofunction Identities - Solving Trigonometric Equations
How to use cofunction identities to solve trigonometric equations?

Example:
Find a possible acute angle solution.
a) cos(2θ + 16°) = sin(θ + 11°)
b) cot(θ) = tan(θ + π/6)

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Power Reducing Formulas - Trigonometric Identities
How to use power reducing formulas to simplify trigonometric expressions?
It contains the power reducing trigonometric identities for sine, cosine, and tangent.

Examples:
sin⁴(x)
sin²(x)cos²(x)
sin⁴(x)cos²(x).

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Half Angle Identities to Evaluate Trigonometric Expressions
Find the exact value of the following:
tan(105°)

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How to use Half Angle Identities to Evaluate Trigonometric Expressions
Find the exact value of the following:
sin(a/2) if cos a = 3/5 for 0° ≤ a ≤ 90°

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