(cos/sin+1/sin)²=1+cos/1-cos
Answers
Modifying just the left-hand side:
We can use the Pythagorean Identity to rewrite
sin
2
x
. The Pythagorean Identity states that
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
∣
∣
a
a
sin
2
x
+
cos
2
x
=
1
a
a
∣
∣
−−−−−−−−−−−−−−−−−−−−−
Which can be rearranged to say that
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
∣
∣
a
a
sin
2
x
=
1
−
cos
2
x
a
a
∣
∣
−−−−−−−−−−−−−−−−−−−−−
Thus, we see that
sin
2
x
1
−
cos
x
=
1
−
cos
2
x
1
−
cos
x
We can now factor the numerator of this fraction.
1
−
cos
2
x
is a difference of squares, which can be factored as:
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
∣
∣
a
a
a
2
−
b
2
=
(
a
+
b
)
(
a
−
b
)
a
a
∣
∣
−−−−−−−−−−−−−−−−−−−−−−−−−−
This can be applied to
1
−
cos
2
x
as follows:
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
∣
∣
a
a
1
−
cos
2
x
=
1
2
−
(
cos
x
)
2
=
(
1
+
cos
x
)
(
1
−
cos
x
)
a
a
∣
∣
−−−−−−−−−−−−−−−−−−−−−−−−−
Therefore,
1
−
cos
2
x
1
−
cos
x
=
(
1
+
cos
x
)
(
1
−
cos
x
)
1
−
cos
x
Since there is
1
−
cos
x
present in both the numerator and denominator, it can be cancelled:
(
1
+
cos
x
)
(
1
−
cos
x
)
1
−
cos
x
=
(
1
+
cos
x
)
(
1
−
cos
x
)
(
1
−
cos
x
)
=
1
+
cos
x
This is what we initially set out to prove.