prove that identity sin 2 A +cos2 A =1
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Answer:
Consider a right angled triangle with an internal angle
θ
:
enter image source here
Then:
sin
θ
=
a
c
cos
θ
=
b
c
So:
sin
2
θ
+
cos
2
θ
=
a
2
c
2
+
b
2
c
2
=
a
2
+
b
2
c
2
By Pythagoras
a
2
+
b
2
=
c
2
, so
a
2
+
b
2
c
2
=
1
So given Pythagoras, that proves the identity for
θ
∈
(
0
,
π
2
)
For angles outside that range we can use:
sin
(
θ
+
π
)
=
−
sin
(
θ
)
cos
(
θ
+
π
)
=
−
cos
(
θ
)
sin
(
−
θ
)
=
−
sin
(
θ
)
cos
(
−
θ
)
=
cos
(
θ
)
So for example:
sin
2
(
θ
+
π
)
+
cos
2
(
θ
+
π
)
=
(
−
sin
θ
)
2
+
(
−
cos
θ
)
2
=
sin
2
θ
+
cos
2
θ
=
1
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