Name: Solutions
Quiz 2: B
1. (10 pts) Verify the following identity.
cos(−θ + β) sec(θ) csc(β) = − cot(−β) − tan(−θ)
Solution: Starting from the left side and working to the right we have that
cos(−θ + β) sec(θ) csc(β) = (cos(−θ) cos(β) − sin(−θ) sin(β))
1
cos(θ) sin(β)
by the angle sum id for cosine and basic Ids
=
cos(θ) cos(β)
sin(θ) sin(β)
+
cos(θ) sin(β)
cos(θ) sin(β)
by odd and even functions and simplifying
= cot(β) + tan(θ)
by simplifying and using basic IDs
= − cot(−β) − tan(−θ)
by odd function properties as desired.
2. (5 pts) Use the angle sum identity for cosine to evaluate
(
cos
Solution: Since cos
( 7π )
12
= cos
(π
we have that cos
12
)
)
+ π3 we have that by the angle sum ID for cosine,
( )
(π )
(π)
(π)
(π)
7π
cos
= cos
cos
− sin
sin
12
4
3
4
3
4
and since
( 7π )
7π
12
√
1 3
1 1
−√
= √
22
2 2
1
3. (5 pts) Use the tangent identity tan(α + β) =
tan(α) + tan(β)
to evaluate
1 − tan(α) tan(β)
(
tan
Solution: Since tan
( 17π )
12
(
= tan
5π π
−
3
4
17π
12
)
)
we have that
(
(
tan
17π
12
)
)
(π)
5π
− tan
3
4
( )
=
(π)
5π
tan
1 + tan
3
4
tan
and since
we have that
(
tan
17π
12
)
2
=
√
− 3−1
√
1− 3
(
)
E.C.: (5 pts) Extra Credit: Prove the graph identity
for sine,
sin π2 −(θ = cos(θ)
(
(
))
(
)
π
π)
Solution: Observe that sin π2 − θ = sin − θ −
= − sin θ −
and since the graph of
2
2
sin(θ) is given by
and the graph of − sin(θ) is given by
(
π)
and the graph of − sin θ −
is given by
2
which is the graph of cos(θ) thus the graph ID follows.
3