prove that 4sin27°=(5+√5)^1/2-(3-√5)^1/2
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Step-by-step explanation:
ts take
(sin27
o
+cos27
o
)
2
=sin
2
27
o
+cos
2
27
o
+2sin27
o
cos27
o
=1+sin54
o
(using identity sin2θ=2sinθcosθ)
⇒(sin27
o
+cos27
o
)
2
=1+sin54
o
=1+
4
5
+1
=
4
1
(5+
5
)
⇒sin27
o
+cos27
o
=
2
1
5+
5
...(1)
Similarly
(sin27
o
−cos27
o
)
2
=1−cos36
o
=1−
4
5
+1
⇒sin27
o
−cos27
o
=
2
1
3−
5
...(2)
From (1) and (2)
4sin27
o
=
5+
5
−
3−
5
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