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Sum and Difference Identities

Sine, cosine, or tangent of two angles that are added or subtracted.

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Sum and Difference Identities

With your knowledge of special angles like the sine and cosine of  and , you can find the sine and cosine of , the difference of  and , and , the sum of  and .  Using what you know about the unit circle and the sum and difference identities, how do you determine  and

Watch This

http://www.youtube.com/watch?v=H-0jQTzfkWQ James Sousa: Sum and Difference Identities for Cosine

http://www.youtube.com/watch?v=hiNDDQyee2E James Sousa: Sum and Difference Identities for Sine

http://www.youtube.com/watch?v=OQP78bwYcWw James Sousa: Sum and Difference Identities for Tangent


There are some intuitive but incorrect formulas for sums and differences with respect to trigonometric functions.  The form below does not work for any trigonometric function and is one of the most common incorrect guesses as to the sum and difference identity.

First look at the derivation of the cosine difference identity:

Start by drawing two arbitrary angles  and .  In the image above  is the angle in red and  is the angle in blue.  The difference  is noted in black as .  The reason why there are two pictures is because the image on the right has the same angle  in a rotated position.  This will be useful to work with because the length of  will be the same as the length of .

The proofs for sine and tangent are left to examples and exercises.  They are listed here for your reference.  Cotangent, secant and cosecant are excluded because you can use reciprocal identities to get those once you have sine, cosine and tangent. 


Example A

Prove the cosine of a sum identity. 

Solution:  Start with the cosine of a difference and make a substitution.  Then use the odd-even identity.


Example B

Find the exact value of  without using a calculator.


A final solution will not have a radical in the denominator.  In this case multiplying through by the conjugate of the denominator will eliminate the radical.  This technique is very common in PreCalculus and Calculus.

Example C

Evaluate the expression exactly without using a calculator.

Solution:  Once you know the general form of the sum and difference identities then you will recognize this as cosine of a difference.

Concept Problem Revisited

In order to evaluate  and  exactly without a calculator, you need to use the sine of a difference and sine of a sum.


The Greek letters used in this concept refer to unknown angles.  They are -alpha, -beta, -theta, -gamma

The symbol  is short hand for “plus or minus.”  The symbol  is shorthand for “minus or plus.”  The order is important because for cosine of a sum, the negative sign is used on the other side of the identity.  This is the opposite of sine of a sum, where a positive sign is used on the other side of the identity. 

Guided Practice

1. Prove the sine of a difference identity. 

2. Use a sum or difference identity to find an exact value of .

3. Prove the following identity:


1. Start with the cofunction identity and then distribute and work out the cosine of a sum and cofunction identities.

2.Start with the definition of cotangent as the inverse of tangent. 

3. Here are the steps:


Find the exact value for each expression by using a sum or difference identity.







7. Prove the sine of a sum identity.

8. Prove the tangent of a sum identity.

9. Prove the tangent of a difference identity.

10. Evaluate without a calculator: .

11. Evaluate without a calculator: .

12. Evaluate without a calculator: .

13. If , then what does  equal?

14. Prove that .

15. Prove that .




The symbol \mp is shorthand for “minus or plus.”


The symbol \pm is shorthand for “plus or minus.”


An identity is a mathematical sentence involving the symbol “=” that is always true for variables within the domains of the expressions on either side.

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