Prove the following identity (sin3−cos3)/− =1+ sin .
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Answer:
Expansion of a cubic
a
3
+
b
3
=
(
a
+
b
)
(
a
2
−
a
b
+
b
2
)
sin
3
x
+
cos
3
x
sin
x
+
cos
x
=
(
sin
x
+
cos
x
)
(
sin
2
x
−
sin
x
cos
x
+
cos
2
x
)
sin
x
+
cos
x
=
sin
2
x
−
sin
x
cos
x
+
cos
2
x
Identity:
sin
2
x
+
cos
2
x
=
1
=
sin
2
x
+
cos
2
x
−
sin
x
cos
x
=
1
−
sin
x
cos
x
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