Sin^2 theta = cos^2 theta
Proof it
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Step-by-step explanation:
sin
2
(
x
)
+
cos
2
(
x
)
=
1
is one of the easier identities to prove using other methods, and so is generally done so.
Still, be all that as it may, let's do a proof using the angle addition formula for cosine:
cos
(
α
+
β
)
=
cos
(
α
)
cos
(
β
)
−
sin
(
α
)
sin
(
β
)
(A proof of the above formula may be found here )
Also, note that sine is an odd function and cosine is an even function, meaning
sin
(
−
x
)
=
−
sin
(
x
)
and
cos
(
−
x
)
=
cos
(
x
)
Now we proceed to the proof.
Let
α
=
x
and
β
=
−
x
⇒
cos
(
x
+
(
−
x
)
)
=
cos
(
x
)
cos
(
−
x
)
−
sin
(
x
)
sin
(
−
x
)
⇒
cos
(
0
)
=
cos
(
x
)
cos
(
x
)
−
(
−
sin
(
x
)
sin
(
x
)
)
∴
1
=
cos
2
(
x
)
+
sin
2
(
x
)
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