Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[LHS = \sin5x\]
\[ = \sin\left( 3x + 2x \right)\]
\[ = \sin3x \times \cos2x + \cos3x \times \sin2x\]
\[ = \left( 3\text{ sin } x - 4 \sin^3 x \right)\left( 2 \cos^2 x - 1 \right) + \left( 4 \cos^3 x - 3\text{ cos } x \right) \times 2\text{ sin } x \text{ cos } x\]
\[ = - 3\text{ sin } x + 4 \sin^3 x + 6\text{ sin } x \cos^2 x - 8 \sin^3 x \cos^2 x + 8\text{ sin } x \cos^4 x - 6\text{ sin } x \cos^2 x\]
\[ = 8\text{ sin } x \cos^4 x - 8 \sin^3 x \cos^2 x - 3\text{ sin } x + 4 \sin^3 x\]
\[ = 5\text{ sin } x \cos^4 x - 10 \sin^3 x \cos^2 x - 3\text{ sin } x + 3\text{ sin } x \cos^4 x + 4 \sin^3 x + 2 \sin^3 x \cos^2 x\]
\[ = 5\text{ sin } x \cos^4 x - 10 \sin^3 x \cos^2 x - 3\text{ sin} x\left( 1 - \cos^4 x \right) + 2 \sin^3 x\left( 2 + \cos^2 x \right)\]
\[= 5\text{ sin } x \cos^4 x - 10 \sin^3 x \cos^2 x - 3\text{ sin }x\left( 1 - \cos^2 x \right)\left( 1 + \cos^2 x \right) + 2 \sin^3 x\left( 2 + \cos^2 x \right)\]
\[ = 5\text{ sin } x \cos^4 x - 10 \sin^3 x \cos^2 x - 3 \sin^3 x\left( 1 + \cos^2 x \right) + 2 \sin^3 x\left( 2 + \cos^2 x \right)\]
\[ = 5\text{ sin } x \cos^4 x - 10 \sin^3 x \cos^2 x - \sin^3 x\left[ 3\left( 1 + \cos^2 x \right) - 2\left( 2 + \cos^2 x \right) \right]\]
\[ = 5\text{ sin } x \cos^4 x - 10 \sin^3 x \cos^2 x - \sin^3 x\left[ 3 + 3 \cos^2 x - 4 - 2 \cos^2 x \right]\]
\[ = 5\text{ sin }x \cos^4 x - 10 \sin^3 x \cos^2 x - \sin^3 x \left[ \cos^2 x - 1 \right]\]
\[ = 5\text{ sin } x \cos^4 x - 10 \sin^3 x \cos^2 x - \sin^3 x \times \left( - \sin^2 x \right)\]
\[ = 5\text{ sin } x \cos^4 x - 10 \sin^3 x \cos^2 x + \sin^5 x\]
\[ = 5 \cos^4 x \text{ sin } x - 10 \cos^2 x \sin^3 x + \sin^5 x\]
\[ = RHS\]
\[\text{ Hence proved } .\]
APPEARS IN
संबंधित प्रश्न
Prove that: \[\sin^2 \left( \frac{\pi}{8} + \frac{x}{2} \right) - \sin^2 \left( \frac{\pi}{8} - \frac{x}{2} \right) = \frac{1}{\sqrt{2}} \sin x\]
Prove that: \[\cos^2 \left( \frac{\pi}{4} - x \right) - \sin^2 \left( \frac{\pi}{4} - x \right) = \sin 2x\]
If \[\cos x = - \frac{3}{5}\] and x lies in IInd quadrant, find the values of sin 2x and \[\sin\frac{x}{2}\] .
If 0 ≤ x ≤ π and x lies in the IInd quadrant such that \[\sin x = \frac{1}{4}\]. Find the values of \[\cos\frac{x}{2}, \sin\frac{x}{2} \text{ and } \tan\frac{x}{2}\]
Prove that: \[\cos 7° \cos 14° \cos 28° \cos 56°= \frac{\sin 68°}{16 \cos 83°}\]
Prove that: \[\cos\frac{\pi}{5}\cos\frac{2\pi}{5}\cos\frac{4\pi}{5}\cos\frac{8\pi}{5} = \frac{- 1}{16}\]
If \[2 \tan \alpha = 3 \tan \beta,\] prove that \[\tan \left( \alpha - \beta \right) = \frac{\sin 2\beta}{5 - \cos 2\beta}\] .
If \[\sin \alpha + \sin \beta = a \text{ and } \cos \alpha + \cos \beta = b\] , prove that
(i)\[\sin \left( \alpha + \beta \right) = \frac{2ab}{a^2 + b^2}\]
If \[\sin \alpha + \sin \beta = a \text{ and } \cos \alpha + \cos \beta = b\] , prove that
(ii) \[\cos \left( \alpha - \beta \right) = \frac{a^2 + b^2 - 2}{2}\]
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
(i) \[\tan\alpha + \tan\beta = \frac{2b}{a + c}\]
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 `tan x + tan (π/3 + x) - tan(π/3 - x) = 3tan 3x`
Prove that: \[\sin^2 \frac{2\pi}{5} - \sin^{2 -} \frac{\pi}{3} = \frac{\sqrt{5} - 1}{8}\]
Prove that: \[\cos 78° \cos 42° \cos 36° = \frac{1}{8}\]
Prove that: \[\cos\frac{\pi}{15}\cos\frac{2\pi}{15}\cos\frac{4\pi}{15}\cos\frac{7\pi}{15} = \frac{1}{16}\]
If \[\frac{\pi}{2} < x < \pi,\] the write the value of \[\sqrt{2 + \sqrt{2 + 2 \cos 2x}}\] in the simplest form.
If \[\pi < x < \frac{3\pi}{2}\], then write the value of \[\sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}}\] .
In a right angled triangle ABC, write the value of sin2 A + Sin2 B + Sin2 C.
If \[5 \sin \alpha = 3 \sin \left( \alpha + 2 \beta \right) \neq 0\] , then \[\tan \left( \alpha + \beta \right)\] is equal to
If \[\tan \left( \pi/4 + x \right) + \tan \left( \pi/4 - x \right) = \lambda \sec 2x, \text{ then } \]
The value of \[\cos^2 \left( \frac{\pi}{6} + x \right) - \sin^2 \left( \frac{\pi}{6} - x \right)\] is
The value of \[2 \sin^2 B + 4 \cos \left( A + B \right) \sin A \sin B + \cos 2 \left( A + B \right)\] is
\[2 \left( 1 - 2 \sin^2 7x \right) \sin 3x\] is equal to
If \[\tan \frac{x}{2} = \frac{\sqrt{1 - e}}{1 + e} \tan \frac{\alpha}{2}\] , then \[\cos \alpha =\]
If \[\left( 2^n + 1 \right) x = \pi,\] then \[2^n \cos x \cos 2x \cos 2^2 x . . . \cos 2^{n - 1} x = 1\]
The value of \[\cos^4 x + \sin^4 x - 6 \cos^2 x \sin^2 x\] is
The value of \[\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right)\] is
The value of \[\tan x + \tan \left( \frac{\pi}{3} + x \right) + \tan \left( \frac{2\pi}{3} + x \right)\] is
If \[n = 1, 2, 3, . . . , \text{ then } \cos \alpha \cos 2 \alpha \cos 4 \alpha . . . \cos 2^{n - 1} \alpha\] is equal to
If \[\text{ tan } x = \frac{a}{b}\], then \[b \cos 2x + a \sin 2x\]
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 ______.
If acos2θ + bsin2θ = c has α and β as its roots, then prove that tanα + tanβ = `(2b)/(a + c)`.
`["Hint: Use the identities" cos2theta = (1 - tan^2theta)/(1 + tan^2theta) "and" sin2theta = (2tantheta)/(1 + tan^2theta)]`.
Find the value of the expression `cos^4 pi/8 + cos^4 (3pi)/8 + cos^4 (5pi)/8 + cos^4 (7pi)/8`
[Hint: Simplify the expression to `2(cos^4 pi/8 + cos^4 (3pi)/8) = 2[(cos^2 pi/8 + cos^2 (3pi)/8)^2 - 2cos^2 pi/8 cos^2 (3pi)/8]`
The value of `(sin 50^circ)/(sin 130^circ)` is ______.
