Advertisements
Advertisements
प्रश्न
Advertisements
उत्तर
\[\frac{\pi}{3} = 60°\]
\[LHS = \text{ cot } x + \cot\left( 60° + x \right) - \cot\left( 60° - x \right)\]
\[ = \frac{1}{\text{ tan } x} + \frac{1}{\tan\left( 60° + x \right)} - \frac{1}{\tan\left( 60° - x \right)}\]
\[= \frac{1}{\text{ tan } x} + \frac{1 - \sqrt{3}\text{ tan } x}{\sqrt{3} + \text{ tan } x} - \frac{1 + \sqrt{3}\text{ tan } x}{\sqrt{3} - \text{ tan } x}\]
\[ \left[ \tan\left( x + y \right) = \frac{\text{ tan } x + \text{ tan } y}{1 - \text{ tan } x \text{ tan } y} \text{ and } \tan\left( x - y \right) = \frac{\text{ tan } x - \text{ tan } y}{1 + \text{ tan } x \text{ tan } y} \right]\]
\[ = \frac{1}{\text{ tan } x} - \frac{8\text{ tan } x}{3 - \tan^2 x}\]
\[ = \frac{3 - \tan^2 x - 8 \tan^2 x}{\left( \text{ tan } x \right)\left( 3 - \tan^2 x \right)}\]
\[ = \frac{3 - 9 \tan^2 x}{3\text{ tan } x - \tan^3 x}\]
\[ = 3\left( \frac{1 - 3 \tan^2 x}{3\text{ tan } x - \tan^3 x} \right)\]
\[ = 3 \times \frac{1}{\tan3x} \left( \because \tan3\theta = \frac{1 - 3 \tan^2 \theta}{3tan\theta - \tan^3 \theta} \right)\]
\[ = 3\cot3x\]
\[ = RHS\]
\[\text{ Hence proved } .\]
APPEARS IN
संबंधित प्रश्न
Prove that: \[\frac{\cos x}{1 - \sin x} = \tan \left( \frac{\pi}{4} + \frac{x}{2} \right)\]
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 \[\sin 3x + \sin 2x - \sin x = 4 \sin x \cos\frac{x}{2} \cos\frac{3x}{2}\]
If \[\text{ tan } x = \frac{b}{a}\] , then find the value of \[\sqrt{\frac{a + b}{a - b}} + \sqrt{\frac{a - b}{a + b}}\] .
Prove that: \[\cos 7° \cos 14° \cos 28° \cos 56°= \frac{\sin 68°}{16 \cos 83°}\]
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 \[\sin \alpha + \sin \beta = a \text{ and } \cos \alpha + \cos \beta = b\] , prove that
(ii) \[\cos \left( \alpha - \beta \right) = \frac{a^2 + b^2 - 2}{2}\]
If \[2 \tan\frac{\alpha}{2} = \tan\frac{\beta}{2}\] , 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) \[\tan\alpha + \tan\beta = \frac{2b}{a + c}\]
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: \[4 \left( \cos^3 10 °+ \sin^3 20° \right) = 3 \left( \cos 10°+ \sin 2° \right)\]
Prove that: \[\cos^3 x \sin 3x + \sin^3 x \cos 3x = \frac{3}{4} \sin 4x\]
Prove that `tan x + tan (π/3 + x) - tan(π/3 - x) = 3tan 3x`
\[\tan x + \tan\left( \frac{\pi}{3} + x \right) - \tan\left( \frac{\pi}{3} - x \right) = 3 \tan 3x\]
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: \[\cos\frac{\pi}{15}\cos\frac{2\pi}{15}\cos\frac{4\pi}{15}\cos\frac{7\pi}{15} = \frac{1}{16}\]
Prove that: \[\cos 6° \cos 42° \cos 66° \cos 78° = \frac{1}{16}\]
In a right angled triangle ABC, write the value of sin2 A + Sin2 B + Sin2 C.
Write the value of \[\cos\frac{\pi}{7} \cos\frac{2\pi}{7} \cos\frac{4\pi}{7} .\]
The value of \[2 \tan \frac{\pi}{10} + 3 \sec \frac{\pi}{10} - 4 \cos \frac{\pi}{10}\] is
If \[\tan \alpha = \frac{1 - \cos \beta}{\sin \beta}\] , then
\[2 \text{ cos } x - \ cos 3x - \cos 5x - 16 \cos^3 x \sin^2 x\]
If \[A = 2 \sin^2 x - \cos 2x\] , then A lies in the interval
\[2 \left( 1 - 2 \sin^2 7x \right) \sin 3x\] is equal to
If \[\tan \frac{x}{2} = \frac{\sqrt{1 - e}}{1 + e} \tan \frac{\alpha}{2}\] , then \[\cos \alpha =\]
The value of \[\tan x + \tan \left( \frac{\pi}{3} + x \right) + \tan \left( \frac{2\pi}{3} + x \right)\] is
If \[\text{ tan } x = \frac{a}{b}\], then \[b \cos 2x + a \sin 2x\]
If \[\tan\alpha = \frac{1}{7}, \tan\beta = \frac{1}{3}\], then
\[\cos2\alpha\] is equal to
The value of `cos^2 48^@ - sin^2 12^@` 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 ______.
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 `(1 - tan^2 15^circ)/(1 + tan^2 15^circ)` is ______.
The value of cos248° – sin212° is ______.
[Hint: Use cos2A – sin2 B = cos(A + B) cos(A – B)]
