Advertisements
Advertisements
प्रश्न
\[\tan x + \tan\left( \frac{\pi}{3} + x \right) - \tan\left( \frac{\pi}{3} - x \right) = 3 \tan 3x\]
उत्तर
\[\frac{\pi}{3} = 60°\]
\[LHS = \text{ tan } x + \tan\left( 60° + x \right) - \tan\left( 60° - x \right)\]
\[ = \text{ tan } x + \left( \frac{\tan60° + \text { tan } x}{1 - \tan60° \text{ tan } x} \right) - \left( \frac{\tan60° - \text{ tan } x}{1 + \tan60° \text{ tan } x} \right)\]
\[ \left[ \tan\left( x + y \right) = \frac{\text{ tan } x + \text{ tan } y}{1 - \text{ tan } x \text{ tan } y} \text{ and } \tan\left( x - y \right) = \frac{\text{ tan } x - \text{ tan } y}{1 + \text{ tan } x \text{ tan } y} \right]\]
\[ = \text{ tanx } + \frac{\sqrt{3} + \text{ tan } x}{1 - \sqrt{3} \text{ tan } x} - \frac{\sqrt{3} - \text{ tan } x}{1 + \sqrt{3} \text{ tan } x}\]
\[ = \text{ tan } x + \frac{\sqrt{3} + 3\text{ tan } x + \text{ tan } x + \sqrt{3} \tan^2 x + \sqrt{3} + 3\text{ tan } x + \text{ tan } x - \sqrt{3} \tan^2 x}{\left( 1 - \sqrt{3} \text{ tan } x \right)\left( 1 + \sqrt{3} \text{ tan } x \right)}\]
\[ = \text{ tan } x + \frac{8\text{ tan } x}{1 - 3 \tan^2 x}\]
\[= \frac{\text{ tan } x - 3 \tan^3 x + 8\text{ tan } x}{1 - 3 \tan^2 x}\]
\[ = \frac{9\text{ tan } x - 3 \tan^3 x}{1 - 3 \tan^2 x}\]
\[ = 3\left( \frac{3\text{ tan } x - \tan^3 x}{1 - 3 \tan^2 x} \right) \left( \because \tan3\theta = \frac{3tan\theta - \tan^3 \theta}{1 - 3 \tan^2 \theta} \right) \]
\[ = 3\tan3x\]
\[ = RHS\]
\[\text{ Hence proved .} \]
APPEARS IN
संबंधित प्रश्न
Prove that: \[\sqrt{2 + \sqrt{2 + 2 \cos 4x}} = 2 \text{ cos } x\]
Prove that: \[\frac{\sin x + \sin 2x}{1 + \cos x + \cos 2x} = \tan x\]
Prove that: \[\cos^2 \frac{\pi}{8} + \cos^2 \frac{3\pi}{8} + \cos^2 \frac{5\pi}{8} + \cos^2 \frac{7\pi}{8} = 2\]
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: \[\sin 4x = 4 \sin x \cos^3 x - 4 \cos x \sin^3 x\]
Prove that:\[\tan\left( \frac{\pi}{4} + x \right) + \tan\left( \frac{\pi}{4} - x \right) = 2 \sec 2x\]
If \[\sin x = \frac{\sqrt{5}}{3}\] and x lies in IInd quadrant, find the values of \[\cos\frac{x}{2}, \sin\frac{x}{2} \text{ and } \tan \frac{x}{2}\] .
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 \[\tan A = \frac{1}{7}\] and \[\tan B = \frac{1}{3}\] , show that cos 2A = sin 4B
Prove that: \[\cos\frac{2\pi}{15} \cos\frac{4\pi}{15} \cos \frac{8\pi}{15} \cos \frac{16\pi}{15} = \frac{1}{16}\]
If \[2 \tan \alpha = 3 \tan \beta,\] prove that \[\tan \left( \alpha - \beta \right) = \frac{\sin 2\beta}{5 - \cos 2\beta}\] .
If \[a \cos2x + b \sin2x = c\] 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 `tan x + tan (π/3 + x) - tan(π/3 - x) = 3tan 3x`
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 \[\left| \cos x \cos \left( \frac{\pi}{3} - x \right) \cos \left( \frac{\pi}{3} + x \right) \right| \leq \frac{1}{4}\] for all values of x
Prove that: \[\sin^2 42° - \cos^2 78 = \frac{\sqrt{5} + 1}{8}\]
If \[\tan\frac{x}{2} = \frac{m}{n}\] , then write the value of m sin x + n cos x.
Write the value of \[\cos^2 76° + \cos^2 16° - \cos 76° \cos 16°\]
If in a \[∆ ABC, \tan A + \tan B + \tan C = 0\], then
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 \[\lambda =\]
If \[\tan \alpha = \frac{1 - \cos \beta}{\sin \beta}\] , then
\[\sin^2 \left( \frac{\pi}{18} \right) + \sin^2 \left( \frac{\pi}{9} \right) + \sin^2 \left( \frac{7\pi}{18} \right) + \sin^2 \left( \frac{4\pi}{9} \right) =\]
If \[A = 2 \sin^2 x - \cos 2x\] , then A lies in the interval
The value of \[\frac{2\left( \sin 2x + 2 \cos^2 x - 1 \right)}{\cos x - \sin x - \cos 3x + \sin 3x}\] is
If α and β are acute angles satisfying \[\cos 2 \alpha = \frac{3 \cos 2 \beta - 1}{3 - \cos 2 \beta}\] , then tan α =
If \[\tan \frac{x}{2} = \frac{\sqrt{1 - e}}{1 + e} \tan \frac{\alpha}{2}\] , then \[\cos \alpha =\]
If \[\text{ tan } x = \frac{a}{b}\], then \[b \cos 2x + a \sin 2x\]
Prove that sin 4A = 4sinA cos3A – 4 cosA sin3A
If tanθ = `1/2` and tanΦ = `1/3`, then the value of θ + Φ is ______.
If sinθ = `(-4)/5` and θ lies in the third quadrant then the value of `cos theta/2` is ______.
If tanA = `(1 - cos "B")/sin"B"`, then tan2A = ______.
