Prove that, tan theta + 2tan 2theta + 4cot 4theta = cot theta
Answers
Answer:
Steve M
Mar 9, 2018
We seek to prove that:
tan
θ
+
2
tan
2
θ
+
4
cot
4
θ
≡
cot
θ
Let us start with the LHS, as this is the most complex part of the expression:
L
H
S
=
tan
θ
+
2
tan
2
θ
+
4
cot
4
θ
=
tan
θ
+
2
tan
2
θ
+
4
tan
4
θ
We can use the tangent double angle formula:
tan
2
A
≡
2
tan
A
1
−
tan
2
A
So then:
L
H
S
=
tan
θ
+
2
tan
2
θ
+
4
tan
2
(
2
θ
)
=
tan
θ
+
2
tan
2
θ
+
4
1
−
tan
2
2
θ
2
tan
2
θ
=
tan
θ
+
2
tan
2
θ
+
2
1
−
tan
2
2
θ
tan
2
θ
=
tan
θ
tan
2
θ
+
2
tan
2
2
θ
+
2
−
2
tan
2
2
θ
tan
2
θ
=
tan
θ
tan
2
θ
+
2
tan
2
θ
=
2
tan
2
θ
1
−
tan
2
θ
+
2
2
tan
θ
1
−
tan
2
θ
=
2
tan
2
θ
+
2
(
1
−
tan
2
θ
)
1
−
tan
2
θ
2
tan
θ
1
−
tan
2
θ
=
2
tan
2
θ
+
2
−
2
tan
2
θ
1
−
tan
2
θ
2
tan
θ
1
−
tan
2
θ
=
2
2
tan
θ
=
1
tan
θ
=
cot
θ
=
R
H
S
QED
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