Advertisements
Advertisements
प्रश्न
Prove that `tan x + tan (π/3 + x) - tan(π/3 - x) = 3tan 3x`
Advertisements
उत्तर
`tan x + tan (π/3 + x) - tan(π/3 - x) = 3tan 3x`
LHS = `tan x + tan (π/3 + x) - tan(π/3 - x)`
`"LHS" = tan x + ((tan (π/3) + tan x)/(1 - tan x tan (π/3))) - ((tan (π/3) - tan x)/(1 + tan x tan (π/3)))`
We know that,
`tan (A + B) = ((tan A + tan B)/(1 - tan A tan B))` and
`tan (A _ B) = ((tan A - tan B)/(1 + tan A tan B))`
So,
`"LHS" = tan x + ((sqrt3 + tan x)/(1 - sqrt3 tan x)) - ((sqrt3 - tan x)/(1 + sqrt3 tan x))`
`"LHS" = tan x + (((1 + sqrt3 tan x)(sqrt3 + tan x) - (1 - sqrt3 tan x)(sqrt3 - tan x))/((1 - sqrt3tan x)(1 + sqrt3 tanx)))`
Simplify and cancel the similar terms of different sign in the above expression
we get,
`"LHS" = tan x + ((0 + 6tan x + 2tan x + 0)/(1 - 3tan^2x))`
`"LHS" = tan x + ((8tan x)/(1 - 3tan^2x))`
`"LHS" = (tan x (1 - 3tan^2x) + 8tan x)/(1 - 3tan^2x)`
`"LHS" = (tan x - 3tan^3x + 8tan x)/(1 - 3tan^2x)`
`"LHS" = (9tan x - 3tan^3x)/(1 - 3tan^2x)`
`"LHS" = 3((3tan x - tan^3x)/(1 - 3tan^2x))`
`"LHS" = 3 tan 3x ...{tan 3x = (3tanx - tan^3x)/(1 - 3tan^2x)`
RHS = 3 tan 3x
Hence proved.
APPEARS IN
संबंधित प्रश्न
Prove that: \[\sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}} = \tan x\]
Prove that: \[\frac{\sin 2x}{1 - \cos 2x} = cot x\]
Prove that: \[\sqrt{2 + \sqrt{2 + 2 \cos 4x}} = 2 \text{ cos } x\]
Prove that: \[\frac{1 - \cos 2x + \sin 2x}{1 + \cos 2x + \sin 2x} = \tan x\]
Prove that: \[\cos^2 \left( \frac{\pi}{4} - x \right) - \sin^2 \left( \frac{\pi}{4} - x \right) = \sin 2x\]
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 \[\sin 3x + \sin 2x - \sin x = 4 \sin x \cos\frac{x}{2} \cos\frac{3x}{2}\]
If \[\cos x = - \frac{3}{5}\] and x lies in the IIIrd quadrant, find the values of \[\cos\frac{x}{2}, \sin\frac{x}{2}, \sin 2x\] .
If \[\sin x = \frac{4}{5}\] and \[0 < x < \frac{\pi}{2}\]
, find the value of sin 4x.
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 \[\sin \alpha + \sin \beta = a \text{ and } \cos \alpha + \cos \beta = b\] , prove that
(i)\[\sin \left( \alpha + \beta \right) = \frac{2ab}{a^2 + b^2}\]
If \[2 \tan\frac{\alpha}{2} = \tan\frac{\beta}{2}\] , prove that \[\cos \alpha = \frac{3 + 5 \cos \beta}{5 + 3 \cos \beta}\]
If \[a \cos2x + b \sin2x = c\] has α and β as its roots, then prove that
(i) \[\tan\alpha + \tan\beta = \frac{2b}{a + c}\]
Prove that: \[\sin 5x = 5 \sin x - 20 \sin^3 x + 16 \sin^5 x\]
\[\cot x + \cot\left( \frac{\pi}{3} + x \right) + \cot\left( \frac{2\pi}{3} + x \right) = 3 \cot 3x\]
Prove that: \[\sin^2 24°- \sin^2 6° = \frac{\sqrt{5} - 1}{8}\]
If \[\cos 4x = 1 + k \sin^2 x \cos^2 x\] , then write the value of k.
If \[\frac{\pi}{2} < x < \pi,\] the write the value of \[\sqrt{2 + \sqrt{2 + 2 \cos 2x}}\] in the simplest form.
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 \[\left|\text { sin } x - \text{ cos } x \right|\] .
The value of \[2 \tan \frac{\pi}{10} + 3 \sec \frac{\pi}{10} - 4 \cos \frac{\pi}{10}\] is
If \[2 \tan \alpha = 3 \tan \beta, \text{ then } \tan \left( \alpha - \beta \right) =\]
If \[\sin \alpha + \sin \beta = a \text{ and } \cos \alpha - \cos \beta = b \text{ then } \tan \frac{\alpha - \beta}{2} =\]
\[\sin^2 \left( \frac{\pi}{18} \right) + \sin^2 \left( \frac{\pi}{9} \right) + \sin^2 \left( \frac{7\pi}{18} \right) + \sin^2 \left( \frac{4\pi}{9} \right) =\]
If \[\tan \left( \pi/4 + x \right) + \tan \left( \pi/4 - x \right) = \lambda \sec 2x, \text{ then } \]
\[\frac{\sin 3x}{1 + 2 \cos 2x}\] is equal to
\[2 \left( 1 - 2 \sin^2 7x \right) \sin 3x\] is equal to
If \[n = 1, 2, 3, . . . , \text{ then } \cos \alpha \cos 2 \alpha \cos 4 \alpha . . . \cos 2^{n - 1} \alpha\] is equal to
Prove that sin 4A = 4sinA cos3A – 4 cosA sin3A
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)]`.
The value of `(1 - tan^2 15^circ)/(1 + tan^2 15^circ)` is ______.
The value of sin50° – sin70° + sin10° is equal to ______.
The value of `sin pi/18 + sin pi/9 + sin (2pi)/9 + sin (5pi)/18` is given by ______.
