If 11x=π,prove that cos x cos 2x cos 3x cos 4x cos 5x =1/32
Answers
Answer:
iven:x=
15
π
\textbf{To prove:}To prove:
cosx\;cos2x\;cos3x\;cos4x\;cos5x\;cos6x\;cos7x=\displaystyle\frac{1}{2^7}cosxcos2xcos3xcos4xcos5xcos6xcos7x=
2
7
1
\textbf{That is, to prove}That is, to prove
cos12^{\circ}\;cos24^{\circ}\;cos36^{\circ}\;cos48^{\circ}\;cos60^{\circ}\;cos72^{\circ}\;cos84^{\circ}=\displaystyle\frac{1}{2^7}cos12
∘
cos24
∘
cos36
∘
cos48
∘
cos60
∘
cos72
∘
cos84
∘
=
2
7
1
\text{Consider,}Consider,
cos12^{\circ}\;cos24^{\circ}\;cos36^{\circ}\;cos48^{\circ}\;cos60^{\circ}\;cos72^{\circ}\;cos84^{\circ}cos12
∘
cos24
∘
cos36
∘
cos48
∘
cos60
∘
cos72
∘
cos84
∘
=cos12^{\circ}\;cos24^{\circ}\;cos36^{\circ}\;cos48^{\circ}\displaystyle(\frac{1}{2})cos72^{\circ}\;cos84^{\circ}=cos12
∘
cos24
∘
cos36
∘
cos48
∘
(
2
1
)cos72
∘
cos84
∘
=\displaystyle(\frac{1}{2})[cos48^{\circ}\;cos12^{\circ}\;cos72^{\circ}][\;cos36^{\circ}\;cos24^{\circ}\;cos84^{\circ}]=(
2
1
)[cos48
∘
cos12
∘
cos72
∘
][cos36
∘
cos24
∘
cos84
∘
]
\text{We know that,}We know that,
\boxed{\bf\,cos(60-A)\;cosA\;cos(60+A)=\frac{1}{4}cos\,3A}}
=\displaystyle(\frac{1}{2})[\frac{1}{4}\;cos3(12^{\circ})][\frac{1}{4}\;cos3(24^{\circ})]=(
2
1
)[
4
1
cos3(12
∘
)][
4
1
cos3(24
∘
)]
=\displaystyle(\frac{1}{2})[\frac{1}{4}\;cos36^{\circ}][\frac{1}{4}\;cos72^{\circ}]=(
2
1
)[
4
1
cos36
∘
][
4
1
cos72
∘
]
=\displaystyle(\frac{1}{32})[cos36^{\circ}][cos72^{\circ}]=(
32
1
)[cos36
∘
][cos72
∘
]
=\displaystyle(\frac{1}{32})[\frac{\sqrt{5}+1}{4}][\frac{\sqrt{5}-1}{4}]=(
32
1
)[
4
5
+1
][
4
5
−1
]
=\displaystyle(\frac{1}{32})[\frac{(\sqrt{5})^2-1^2}{16}]=(
32
1
)[
16
(
5
)
2
−1
2
]
=\displaystyle(\frac{1}{32})[\frac{5-1}{16}]=(
32
1
)[
16
5−1
]
=\displaystyle(\frac{1}{32})[\frac{4}{16}]=(
32
1
)[
16
4
]
=\displaystyle(\frac{1}{32})[\frac{1}{4}]=(
32
1
)[
4
1
]
=\displaystyle(\frac{1}{2^5})[\frac{1}{2^2}]=(
2
5
1
)[
2
2
1
]
=\bf\displaystyle\frac{1}{2^7}=
2
7
1
\therefore\bf\,cos12^{\circ}\;cos24^{\circ}\;cos36^{\circ}\;cos48^{\circ}\;cos60^{\circ}\;cos72^{\circ}\;cos84^{\circ}=\displaystyle\frac{1}{2^7}∴cos12
∘
cos24
∘
cos36
∘
cos48
∘
cos60
∘
cos72
∘
cos84
∘
=
2
7
1
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