Even and Odd Trigonometric Identities

Show your work for each problem.

1.
Which trigonometric function is even?
1. a)Cotangent
2. b)Tangent
3. c)Sine
4. d)Cosine
Answer: Cosine
Cosine is an even function because . The cosine graph is symmetric about the y-axis. Sine, tangent, and cotangent are all odd functions.
2.
What is ?
1. a)
2. b)
3. c)
4. d)
Answer:
Since sine is odd: . We know , so the negative of that is .
3.
If , what is ? (Enter as a fraction with sqrt for square root)
Answer: sqrt(2)/2
Cosine is even, so . The negative angle gives the same result as the positive angle for even functions.
4.
Which of the following is an odd function?
1. a)
2. b)
3. c)
4. d)
Answer:
Tangent is odd because . Cosine and secant are even functions. is the square of an even function, which makes it even.
5.
Simplify to a single expression with . (Include the sign)
Answer: -tan(60)
Since tangent is odd: . The negative sign moves from inside the function to outside.
6.
Simplify:
Answer: -tan(x)
1. Apply the odd identity to : -sin(x)
2. Apply the even identity to : cos(x)
3. Simplify : -tan(x)
7.
If , what is ?
1. a)
2. b)
3. c)
4. d)
Answer:
Since sine is odd: . The sign of the sine value flips when the angle is negated.
8.
Simplify to an expression using only positive angles.
Answer: cos(x) - sin(x)
9.
Simplify:
Answer: -sec(theta) tan(theta)
1. Is secant even or odd? even
2. So sec(theta)
3. Is tangent even or odd? odd
4. So -tan(theta)
5. Multiply: -sec(theta) tan(theta)
10.
What is ?
1. a)
2. b)
3. c)
4. d)
Answer:
Since cotangent is odd: . We know because .
11.
Is the function even, odd, or neither? (Enter: even, odd, or neither)
Answer: odd
. Since , the function is odd.
12.
Prove that for all .
Answer: 0
1. Apply the odd identity to : -sin(x)
2. Substitute into the expression: 0
3. Is this true for all values of ? (yes/no) yes
13.
Simplify:
Answer: -2sin(x)
1. Apply even identity to : cos(x)
2. Apply odd identity to : -tan(x)
3. Apply odd identity to : -sin(x)
4. Rewrite: -cos(x)tan(x) - sin(x)
5. Simplify using : sin(x)
6. Final answer: -2sin(x)
14.
If , is even, odd, or neither?
1. a)Neither
2. b)Cannot be determined
3. c)Even
4. d)Odd
Answer: Even
. Since , the function is even. Squaring an odd function produces an even function!
15.
Prove:
Answer: -tan(x/2)
1. Apply even identity: cos(x)
2. Apply odd identity: -sin(x)
3. Substitute: -((1-cos(x))/sin(x))
4. The half-angle identity states . So our answer is: -tan(x/2)
16.
If and , what is ?
Answer: -0.48