The value of cos 48 degrees cos 12 degrees
Answers
Step-by-step explanation:
Step-by-step explanation:
To find : Solve the expression \cos (48^\circ)\times \cos (12^\circ)cos(48
∘
)×cos(12
∘
)
Solution :
Expression \cos (48^\circ)\times \cos (12^\circ)cos(48
∘
)×cos(12
∘
)
Applying trigonometric identities,
\cos A\cos B=\frac{1}{2}[\cos(A+B)+\cos(A-B)]cosAcosB=
2
1
[cos(A+B)+cos(A−B)]
Substitute, A=48 and B = 12
\cos (48^\circ) \cos (12^\circ)=\frac{1}{2}[\cos(48+12)+\cos(48-12)]cos(48
∘
)cos(12
∘
)=
2
1
[cos(48+12)+cos(48−12)]
\cos (48^\circ) \cos (12^\circ)=\frac{1}{2}[\cos(60)+\cos(36)]cos(48
∘
)cos(12
∘
)=
2
1
[cos(60)+cos(36)]
Substitute, \cos(60)=\frac{1}{2}\ , \cos (36)=\frac{\sqrt5+1}{4}cos(60)=
2
1
,cos(36)=
4
5
+1
\cos (48^\circ) \cos (12^\circ)=\frac{1}{2}[\frac{1}{2}+\frac{\sqrt5+1}{4}]cos(48
∘
)cos(12
∘
)=
2
1
[
2
1
+
4
5
+1
]
\cos (48^\circ) \cos (12^\circ)=\frac{1}{4}+\frac{\sqrt5+1}{8}cos(48
∘
)cos(12
∘
)=
4
1
+
8
5
+1
\cos (48^\circ) \cos (12^\circ)=\frac{2+\sqrt5+1}{8}cos(48
∘
)cos(12
∘
)=
8
2+
5
+1
\cos (48^\circ) \cos (12^\circ)=\frac{3+\sqrt5}{8}cos(48
∘
)cos(12
∘
)=
8
3+
5
Therefore, \cos (48^\circ) \cos (12^\circ)=\frac{3+\sqrt5}{8}cos(48
∘
)cos(12
∘
)=
8
3+
5
cos(48+12)=cos 60
cos 60=1/2
I hope it helps you.
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