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Simplify the expression by rewriting and using identities:
We can start with the Pythagorean identity.
Now we can simplify by substituting for We have
Use algebraic techniques to verify the identity:
(Hint: Multiply the numerator and denominator on the left side by
Access these online resources for additional instruction and practice with the fundamental trigonometric identities.
Pythagorean identities | |
Even-odd identities | |
Reciprocal identities | |
Quotient identities |
We know is an even function, and and are odd functions. What about and Are they even, odd, or neither? Why?
All three functions, and are even.
This is because and
Examine the graph of on the interval How can we tell whether the function is even or odd by only observing the graph of
After examining the reciprocal identity for explain why the function is undefined at certain points.
When then which is undefined.
All of the Pythagorean identities are related. Describe how to manipulate the equations to get from to the other forms.
For the following exercises, use the fundamental identities to fully simplify the expression.
For the following exercises, simplify the first trigonometric expression by writing the simplified form in terms of the second expression.
For the following exercises, verify the identity.
For the following exercises, prove or disprove the identity.
For the following exercises, determine whether the identity is true or false. If false, find an appropriate equivalent expression.
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