prove that (2+√3) ^2-4(2+√3) +1=0
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Answer:
Answer
tan30=
3
1
1−tan
2
15
2tan15
=
3
1
Let tan15=a
2
3
a=1−a
2
a
2
+2
3
a−1=0
a=
2
−2
3
±
12+4
(∵tan15>0)
∴tan15=2−
3
1−tan
2
7
2
1
°
2tan7
2
1
°
=2−
3
Let tan7
2
1
°=b
2b=(2−
3
)(1−b
2
)
(2−
3
)b
2
+2b−(2−
3
)=0
∴b=
2(2−
3
)
−2+
4+4(2−
3
)
2
(∵tan7
2
1
°>0)
=
(2−
3
)
−1+
8−4
3
∴tan7
2
1
°=
2−
3
6
−
2
−1
tan82
2
1
°=cot7
2
1
°=
6
−
2
−1
2−
3
=
(
6
−(
2
+1))(
6
+
2
+1)
(2−
3
)(
6
+
2
+1)
=
6−(
2
+1)
2
2
6
−3
2
+2
2
−
6
+2−
3
=
3−2
2
6
−
2
+2−
3
=
(
2
−1)
2
(
2
−1)(
3
+
2
)
=
2
−1
3
+
2
=(
3
+
2
)(
2
+1)
∴tan82
2
1
°=(
3
+
2
)(
2
+1)
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