Answers
Answer:
The second and third identities can be obtained by manipulating the first. The identity
1
+
cot
2
θ
=
csc
2
θ
is found by rewriting the left side of the equation in terms of sine and cosine.
Prove:
1
+
cot
2
θ
=
csc
2
θ
1
+
cot
2
θ
=
(
1
+
cos
2
θ
sin
2
θ
)
Rewrite the left side
.
=
(
sin
2
θ
sin
2
θ
)
+
(
cos
2
θ
sin
2
θ
)
Write both terms with the common denominator
.
=
sin
2
θ
+
cos
2
θ
sin
2
θ
=
1
sin
2
θ
=
csc
2
θ
Similarly,
1
+
tan
2
θ
=
sec
2
θ
can be obtained by rewriting the left side of this identity in terms of sine and cosine. This gives
1
+
tan
2
θ
=
1
+
(
sin
θ
cos
θ
)
2
Rewrite left side
.
=
(
cos
θ
cos
θ
)
2
+
(
sin
θ
cos
θ
)
2
Write both terms with the common denominator
.
=
cos
2
θ
+
sin
2
θ
cos
2
θ
=
1
cos
2
θ
=
sec
2
θ
The next set of fundamental identities is the set of even-odd identities. The even-odd identities relate the value of a trigonometric function at a given angle to the value of the function at the opposite angle and determine whether the identity is odd or even.