Prove that-2 cos-13cos9 -13-cos3 -13-cos5 -13-0

We have L.H.S. = 2 cos π/13 cos 9π/13+ cos 3π/13+ cos 5π/13 = cos ( 9π/13+π/13 ) + cos (9π/13 π/13 )+ cos3π/13+ cos5π/13 [∵ 2 cos A cos B = cos (A+B) + cos (A ...

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Standard XII

Mathematics

Trigonometric Ratios of Multiples of an Angle

Prove that:2 ...

Question

Prove that:

$2 cos \frac{π}{13} cos \frac{9 π}{13} + cos \frac{3 π}{13} + cos \frac{5 π}{13} = 0$

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Solution

We have L.H.S.

= 2 cos $\frac{π}{13}$ cos $\frac{9 π}{13}$ + cos $\frac{3 π}{13}$ + cos $\frac{5 π}{13}$

= cos $(\frac{9 π}{13} + \frac{π}{13}) + cos (\frac{9 π}{13} - \frac{π}{13}) + cos \frac{3 π}{13} + cos \frac{5 π}{13}$

[ $∵$ 2 cos A cos B = cos (A+B) + cos (A-B)]

= cos $\frac{10 π}{13}$ + cos $\frac{8 π}{13}$ + cos $\frac{3 π}{13}$ + cos $\frac{5 π}{13}$

= cos $(π - \frac{3 π}{13}) + cos (π - \frac{5 π}{13}) + cos \frac{3 π}{13} + cos \frac{5 π}{13}$

[ $∵ cos (π - θ) = - cos θ$ ]

=- cos $\frac{3 π}{13} - cos \frac{5 π}{13} + cos \frac{3 π}{13} + cos \frac{5 π}{13}$ = 0

= R.H.S.

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Similar questions

Q. Prove that

2 cos \frac{π}{13} cos \frac{9 π}{13} + cos \frac{3 π}{13} + cos \frac{5 π}{13} = 0

$2 c o s \frac{π}{13} c o s \frac{9 π}{13} + c o s \frac{3 π}{13} + c o s \frac{5 π}{13} =$

Q. Value of :

2 cos \frac{π}{13} cos \frac{9 π}{13} + cos \frac{3 π}{13} + cos \frac{5 π}{13}

2 cos \frac{π}{13} cos \frac{9 π}{13} + cos \frac{3 π}{13} + cos \frac{5 π}{13}

is equal to

Q. The value of

2 cos \frac{π}{13} cos \frac{9 π}{13} + cos \frac{3 π}{13} + cos \frac{5 π}{13}

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Prove that: 2cosπ13cos9π13+cos3π13+cos5π13=0

Prove that:

$2 cos \frac{π}{13} cos \frac{9 π}{13} + cos \frac{3 π}{13} + cos \frac{5 π}{13} = 0$