(f) The value of cot 15° is
Answers
Answer:
The value of cot 15° is 2+\sqrt{3}2+
3
.
Step-by-step explanation:
The given expression is
\cot(15^{\circ})cot(15
∘
)
It can be written as
\cot(15^{\circ})=\cot(45^{\circ})-30^{\circ})cot(15
∘
)=cot(45
∘
)−30
∘
)
\cot(15^{\circ})=\frac{\cot(45)\cot (30)+1}{\cot(30)-\cot (45)}cot(15
∘
)=
cot(30)−cot(45)
cot(45)cot(30)+1
\cot(15^{\circ})=\frac{(1)(\sqrt{3})+1}{\sqrt{3}-1}cot(15
∘
)=
3
−1
(1)(
3
)+1
\cot(15^{\circ})=\frac{\sqrt{3}+1}{\sqrt{3}-1}cot(15
∘
)=
3
−1
3
+1
Multiply both numerator and denominator by (\sqrt{3}+1)(
3
+1) .
\cot(15^{\circ})=\frac{\sqrt{3}+1}{\sqrt{3}-1}\times \frac{\sqrt{3}+1}{\sqrt{3}+1}cot(15
∘
)=
3
−1
3
+1
×
3
+1
3
+1
\cot(15^{\circ})=\frac{(\sqrt{3}+1)^2}{(\sqrt{3})^2-1^2}cot(15
∘
)=
(
3
)
2
−1
2
(
3
+1)
2
\cot(15^{\circ})=\frac{(3+1+2\sqrt{3}}{3-1}cot(15
∘
)=
3−1
(3+1+2
3
\cot(15^{\circ})=\frac{(2(2+\sqrt{3})}{2}=2+\sqrt{3}cot(15
∘
)=
2
(2(2+
3
)
=2+
3
Therefore the value of cot 15° is 2+\sqrt{3}2+
3
.