Advertisements
Advertisements
प्रश्न
पर्याय
- \[16 \cos^4 x - 12 \cos^2 x + 1\]
- \[16 \cos^4 x + 12 \cos^2 x + 1\]
- \[16 \cos^4 x - 12 \cos^2 x - 1\]
- \[16 \cos^4 x + 12 \cos^2 x - 1\]
Advertisements
उत्तर
\[\text{ To find } : \frac{\sin 5x}{\text{ sin } x}\]
\[\text{ Now} , \]
\[\sin5x = \sin\left( 3x + 2x \right)\]
\[ = \sin3x\cos2x + \cos3x\sin2x\]
\[ = \left( 3\text{ sin } x - 4 \sin^3 x \right)\left( 1 - 2 \sin^2 x \right) + \left( 4 \cos^3 x - 3\text{ cos } x \right)\left( 2\text{ sin } x\text{ cos } x \right)\]
\[ = \left( 3\sin x - 6 \sin^3 x - 4 \sin^3 x + 8 \sin^5 x \right) + 2\text{ sin } x \cos^2 x\left( 4 \cos^2 x - 3 \right)\]
\[ = \left( 3\sin x - 10 \sin^3 x + 8 \sin^5 x \right) + 2\text{ sin } x\left( 1 - \sin^2 x \right)\left[ 4\left( 1 - \sin^2 x \right) - 3 \right]\]
\[ = \left( 3\sin x - 10 \sin^3 x + 8 \sin^5 x \right) + \left( 2\text{ sin } x - 2 \sin^3 x \right)\left( 4 - 4 \sin^2 x - 3 \right)\]
\[ = \left( 3\sin x - 10 \sin^3 x + 8 \sin^5 x \right) + \left( 2\text{ sin } x - 8 \sin^3 x - 2 \sin^3 x + 8 \sin^5 x \right)\]
\[ = 5\text{ sin } x - 20 \sin^3 x + 16 \sin^5 x\]
\[ \therefore \frac{\sin 5x}{\text{ sin } x} = \frac{5\text{ sin } x - 20 \sin^3 x + 16 \sin^5 x}{\text{ sin } x} \]
\[ = 5 - 20 \sin^2 x + 16 \sin^4 x \]
\[ = 5 - 20\left( 1 - \cos^2 x \right) + 16 \left( 1 - \cos^2 x \right)^2 \]
\[ = 5 - 20 + 20 \cos^2 x + 16\left( 1 + \cos^4 x - 2 \cos^2 x \right)\]
\[ = 5 - 20 + 20 \cos^2 x + 16 + 16 \cos^4 x - 32 \cos^2 x\]
\[ = 16 \cos^4 x - 12 \cos^2 x + 1\]
APPEARS IN
संबंधित प्रश्न
Prove that: \[\sqrt{2 + \sqrt{2 + 2 \cos 4x}} = 2 \text{ cos } x\]
Prove that: \[1 + \cos^2 2x = 2 \left( \cos^4 x + \sin^4 x \right)\]
Prove that: \[\sin 4x = 4 \sin x \cos^3 x - 4 \cos x \sin^3 x\]
Show that: \[2 \left( \sin^6 x + \cos^6 x \right) - 3 \left( \sin^4 x + \cos^4 x \right) + 1 = 0\]
Prove that: \[\cos^6 A - \sin^6 A = \cos 2A\left( 1 - \frac{1}{4} \sin^2 2A \right)\]
Prove that: \[\cos^6 A - \sin^6 A = \cos 2A\left( 1 - \frac{1}{4} \sin^2 2A \right)\]
Prove that: \[\cos 4x - \cos 4\alpha = 8 \left( \cos x - \cos \alpha \right) \left( \cos x + \cos \alpha \right) \left( \cos x - \sin \alpha \right) \left( \cos x + \sin \alpha \right)\]
If \[\cos x = - \frac{3}{5}\] and x lies in IInd quadrant, find the values of sin 2x and \[\sin\frac{x}{2}\] .
If \[\sin x = \frac{4}{5}\] and \[0 < x < \frac{\pi}{2}\]
, find the value of sin 4x.
If \[\cos x = \frac{\cos \alpha + \cos \beta}{1 + \cos \alpha \cos \beta}\] , prove that \[\tan\frac{x}{2} = \pm \tan\frac{\alpha}{2}\tan\frac{\beta}{2}\]
If \[a \cos2x + b \sin2x = c\] has α and β as its roots, then prove that
(iii)\[\tan\left( \alpha + \beta \right) = \frac{b}{a}\]
Prove that: \[\sin 5x = 5 \sin x - 20 \sin^3 x + 16 \sin^5 x\]
Prove that \[\left| \sin x \sin \left( \frac{\pi}{3} - x \right) \sin \left( \frac{\pi}{3} + x \right) \right| \leq \frac{1}{4}\] for all values of x
Prove that: \[\cos 36° \cos 42° \cos 60° \cos 78° = \frac{1}{16}\]
Prove that: \[\cos\frac{\pi}{15} \cos \frac{2\pi}{15} \cos \frac{3\pi}{15} \cos \frac{4\pi}{15} \cos \frac{5\pi}{15} \cos\frac{6\pi}{15} \cos \frac{7\pi}{15} = \frac{1}{128}\]
If \[\frac{\pi}{2} < x < \pi,\] the write the value of \[\sqrt{2 + \sqrt{2 + 2 \cos 2x}}\] in the simplest form.
In a right angled triangle ABC, write the value of sin2 A + Sin2 B + Sin2 C.
If \[\text{ tan } A = \frac{1 - \text{ cos } B}{\text{ sin } B}\]
, then find the value of tan2A.
For all real values of x, \[\cot x - 2 \cot 2x\] is equal to
If \[\tan \alpha = \frac{1 - \cos \beta}{\sin \beta}\] , then
\[\sin^2 \left( \frac{\pi}{18} \right) + \sin^2 \left( \frac{\pi}{9} \right) + \sin^2 \left( \frac{7\pi}{18} \right) + \sin^2 \left( \frac{4\pi}{9} \right) =\]
\[2 \text{ cos } x - \ cos 3x - \cos 5x - 16 \cos^3 x \sin^2 x\]
If \[A = 2 \sin^2 x - \cos 2x\] , then A lies in the interval
The value of \[\frac{\cos 3x}{2 \cos 2x - 1}\] is equal to
The value of \[\cos^2 \left( \frac{\pi}{6} + x \right) - \sin^2 \left( \frac{\pi}{6} - x \right)\] is
\[2 \left( 1 - 2 \sin^2 7x \right) \sin 3x\] is equal to
The value of \[\tan x + \tan \left( \frac{\pi}{3} + x \right) + \tan \left( \frac{2\pi}{3} + x \right)\] is
The value of sin 20° sin 40° sin 60° sin 80° is ______.
The value of `cos pi/5 cos (2pi)/5 cos (4pi)/5 cos (8pi)/5` is ______.
The value of `sin pi/10 sin (13pi)/10` is ______.
`["Hint: Use" sin18^circ = (sqrt5 - 1)/4 "and" cos36^circ = (sqrt5 + 1)/4]`
If sinθ = `(-4)/5` and θ lies in the third quadrant then the value of `cos theta/2` is ______.
The value of cos248° – sin212° is ______.
[Hint: Use cos2A – sin2 B = cos(A + B) cos(A – B)]
