Even-Odd Identities

Last Updated : 23 Jul, 2025

Even-odd identities are mathematical relationships that describe how the sine and cosine functions behave based on angle.

  • Even Functions: A function f(x) is called even if f(āˆ’x) = f(x). This means that the function is symmetrical about the y-axis. The cosine function is an example of an even function.
  • Odd Functions: A function f(x) is called odd if f(āˆ’x) = āˆ’f(x). This means that the function is symmetrical about the origin. The sine function is an example of an odd function.
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Even-Odd Identities

In trigonometry, there are many types of identities, including Pythagorean identities, even-odd identities, reciprocal identities, and sum and difference identities.

Even-Odd Identities in Trigonometry

In trigonometry, some functions behave as even functions, while others behave as odd. Hereā€™s how they break down:

Even-Odd IdentitiesEven or Odd
sinā”(āˆ’x) = āˆ’sinā”(x)Odd
cosā”(āˆ’x) = cosā”(x)Even
tanā”(āˆ’x) = āˆ’tanā”(x)Odd
cosecā”(āˆ’x) = āˆ’cosecā”(x)Odd
secā”(āˆ’x) = secā”(x)Even
cotā”(āˆ’x) = āˆ’cotā”(x)Odd

Proof of Even-Odd Identities

As we know, -x (if x < 90Ā°) lies in the fourth quadrant, and only positive values in that quadrant are cos and sec.

  • cosā”(āˆ’x) = cosā”(x) and secā”(āˆ’x) = secā”(x)
Even-Odd-Identities
Unit Circle

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Solved Questions on Even-Odd Identities

Question 1: Evaluate cosā”(āˆ’60Ā°) + sinā”(āˆ’30Ā°).

Solution:

First, simplify each term using the appropriate identities:

  • For cosā”(āˆ’60Ā°): cosā”(āˆ’60Ā°) = cosā”(60Ā°) = 1/2
  • For sinā”(āˆ’30Ā°): sinā”(āˆ’30Ā°) = āˆ’sinā”(30Ā°) = āˆ’1/2ā€‹

Now, combine the results:

cosā”(āˆ’60Ā°) + sinā”(āˆ’30Ā°) = 1/2 + (-1/2) = 1/2 - 1/2 = 0

Question 2: Prove that sinā”(āˆ’x) + sinā”(x) = 0

Solution:

Using the odd identity for sine:
sinā”(āˆ’x) = āˆ’sinā”(x)

Now substitute this into the left side of the equation:
sinā”(āˆ’x) + sinā”(x) = āˆ’sinā”(x) + sinā”(x) = 0

Thus, the identity is proven to be true.

Question 3: Find cosā”(āˆ’135Ā°) and sinā”(āˆ’135Ā°).

Solution:

  • For cosā”(āˆ’135Ā°): cosā”(āˆ’135Ā°) = cosā”(135Ā°) = cosā”(90Ā° + 45Ā°) = - sin(45Ā°) = āˆ’1/āˆš2ā€‹ā€‹ [As cos(90Ā° + x) = -sin x]
  • For sinā”(āˆ’135Ā°): sinā”(āˆ’135Ā°) = āˆ’ sinā”(135Ā°) = -sin(90Ā° + 45Ā°) = -cos(45Ā°) = āˆ’1/āˆš2 [As sin(90Ā° + x) = cos x]

Worksheet: Even-Odd Identities

Worksheet-on-Even-Odd-Identities
Worksheet on even-odd identities

You can download this free worksheet on Even-Odd Identities from below:

Download Free Worksheet on Even-Odd Identities

Conclusion

In conclusion, even-odd identities are important concepts in trigonometry that help us understand how sine and cosine functions behave with negative angles. By recognizing that cosine is an even function and sine is an odd function, we can simplify calculations and solve problems more easily.

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