Advertisements
Advertisements
प्रश्न
Prove that: \[\cos \frac{\pi}{65} \cos \frac{2\pi}{65} \cos\frac{4\pi}{65} \cos\frac{8\pi}{65} \cos\frac{16\pi}{65} \cos\frac{32\pi}{65} = \frac{1}{64}\]
Advertisements
उत्तर
On dividing and multiplying by \[2\sin\frac{\pi}{65}\] , we get
\[= \frac{1}{2\sin\frac{\pi}{65}} \times 2\sin\frac{\pi}{65} \times \cos\frac{\pi}{65} \times \cos\frac{2\pi}{65} \times \cos\frac{4\pi}{65} \times \cos\frac{8\pi}{65} \times \cos\frac{16\pi}{65} \times \cos\frac{32\pi}{65}\]
\[ = \frac{2 \times \sin2\frac{\pi}{65}}{2 \times 2\sin\frac{\pi}{65}} \times \cos\frac{2\pi}{65} \times \cos\frac{4\pi}{65} \times \cos\frac{8\pi}{65} \times \cos\frac{16\pi}{65} \times \cos\frac{32\pi}{65} \]
\[ = \frac{2 \times \sin4\frac{\pi}{65}}{2 \times 4\sin\frac{\pi}{65}} \times \cos\frac{4\pi}{65} \times \cos\frac{8\pi}{65} \times \cos\frac{16\pi}{65} \times \cos\frac{32\pi}{65}\]
\[ = \frac{2 \times \sin8\frac{\pi}{65}}{2 \times 8\sin\frac{\pi}{65}} \times \cos\frac{8\pi}{65} \times \cos\frac{16\pi}{65} \times \cos\frac{32\pi}{65}\]
\[= \frac{2 \times \sin16\frac{\pi}{65}}{2 \times 16\sin\frac{\pi}{65}} \times \cos\frac{16\pi}{65} \times \cos\frac{32\pi}{65}\]
\[ = \frac{2 \times \sin32\frac{\pi}{65}}{2 \times 32\sin\frac{\pi}{65}} \times \cos\frac{32\pi}{65}\]
\[ = \frac{\sin64\frac{\pi}{65}}{64\sin\frac{\pi}{65}} = \frac{\sin\left( \pi - \frac{\pi}{65} \right)}{64\sin\frac{\pi}{65}}\]
\[ = \frac{\sin\frac{\pi}{65}}{64\sin\frac{\pi}{65}} \left[ \because \sin\left( \pi - \theta \right) = sin\theta \right]\]
\[ = \frac{1}{64} = RHS\]
\[\text{ Hence proved } .\]
APPEARS IN
संबंधित प्रश्न
Prove that: \[\frac{\sin 2x}{1 + \cos 2x} = \tan x\]
Prove that: \[\frac{1 - \cos 2x + \sin 2x}{1 + \cos 2x + \sin 2x} = \tan x\]
Prove that: \[\sin^2 \frac{\pi}{8} + \sin^2 \frac{3\pi}{8} + \sin^2 \frac{5\pi}{8} + \sin^2 \frac{7\pi}{8} = 2\]
Prove that: \[1 + \cos^2 2x = 2 \left( \cos^4 x + \sin^4 x \right)\]
Prove that: \[\cos^3 2x + 3 \cos 2x = 4\left( \cos^6 x - \sin^6 x \right)\]
Prove that: \[\cos 4x = 1 - 8 \cos^2 x + 8 \cos^4 x\]
Prove that: \[\sin 4x = 4 \sin x \cos^3 x - 4 \cos x \sin^3 x\]
Show that: \[3 \left( \sin x - \cos x \right)^4 + 6 \left( \sin x + \cos \right)^2 + 4 \left( \sin^6 x + \cos^6 x \right) = 13\]
Show that: \[2 \left( \sin^6 x + \cos^6 x \right) - 3 \left( \sin^4 x + \cos^4 x \right) + 1 = 0\]
Prove that: \[\cot^2 x - \tan^2 x = 4 \cot 2 x \text{ cosec } 2 x\]
If \[\cos x = - \frac{3}{5}\] and x lies in IInd quadrant, find the values of sin 2x and \[\sin\frac{x}{2}\] .
If \[\text{ tan } x = \frac{b}{a}\] , then find the value of \[\sqrt{\frac{a + b}{a - b}} + \sqrt{\frac{a - b}{a + b}}\] .
If \[\sec \left( x + \alpha \right) + \sec \left( x - \alpha \right) = 2 \sec x\] , prove that \[\cos x = \pm \sqrt{2} \cos\frac{\alpha}{2}\]
If \[\cos \alpha + \cos \beta = \frac{1}{3}\] and sin \[\sin\alpha + \sin \beta = \frac{1}{4}\] , prove that \[\cos\frac{\alpha - \beta}{2} = \pm \frac{5}{24}\]
If \[a \cos2x + b \sin2x = c\] has α and β as its roots, then prove that
(ii) \[\tan\alpha \tan\beta = \frac{c - a}{c + a}\]
If \[a \cos2x + b \sin2x = c\] has α and β as its roots, then prove that
(iii)\[\tan\left( \alpha + \beta \right) = \frac{b}{a}\]
If \[\cos\alpha + \cos\beta = 0 = \sin\alpha + \sin\beta\] , then prove that \[\cos2\alpha + \cos2\beta = - 2\cos\left( \alpha + \beta \right)\] .
Prove that \[\left| \cos x \cos \left( \frac{\pi}{3} - x \right) \cos \left( \frac{\pi}{3} + x \right) \right| \leq \frac{1}{4}\] for all values of x
Prove that: \[\cos 36° \cos 42° \cos 60° \cos 78° = \frac{1}{16}\]
If \[\cos 4x = 1 + k \sin^2 x \cos^2 x\] , then write the value of k.
If \[\pi < x < \frac{3\pi}{2}\], then write the value of \[\sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}}\] .
Write the value of \[\cos\frac{\pi}{7} \cos\frac{2\pi}{7} \cos\frac{4\pi}{7} .\]
If \[\text{ tan } A = \frac{1 - \text{ cos } B}{\text{ sin } B}\]
, then find the value of tan2A.
If \[\text{ sin } x + \text{ cos } x = a\], find the value of \[\left|\text { sin } x - \text{ cos } x \right|\] .
If \[\cos x = \frac{1}{2} \left( a + \frac{1}{a} \right),\] and \[\cos 3 x = \lambda \left( a^3 + \frac{1}{a^3} \right)\] then \[\lambda =\]
The value of \[\tan x \sin \left( \frac{\pi}{2} + x \right) \cos \left( \frac{\pi}{2} - x \right)\]
The value of \[\frac{\cos 3x}{2 \cos 2x - 1}\] is equal to
\[2 \left( 1 - 2 \sin^2 7x \right) \sin 3x\] is equal to
If \[\tan x = t\] then \[\tan 2x + \sec 2x =\]
The greatest value of sin x cos x is ______.
The value of `cos pi/5 cos (2pi)/5 cos (4pi)/5 cos (8pi)/5` is ______.
If tanθ + sinθ = m and tanθ – sinθ = n, then prove that m2 – n2 = 4sinθ tanθ
[Hint: m + n = 2tanθ, m – n = 2sinθ, then use m2 – n2 = (m + n)(m – n)]
If tanθ = `1/2` and tanΦ = `1/3`, then the value of θ + Φ is ______.
The value of cos12° + cos84° + cos156° + cos132° is ______.
The value of cos248° – sin212° is ______.
[Hint: Use cos2A – sin2 B = cos(A + B) cos(A – B)]
If k = `sin(pi/18) sin((5pi)/18) sin((7pi)/18)`, then the numerical value of k is ______.
