Advertisements
Advertisements
Question
The value of \[\left( \cot \frac{x}{2} - \tan \frac{x}{2} \right)^2 \left( 1 - 2 \tan x \cot 2 x \right)\] is
Options
1
2
3
4
Advertisements
Solution
4
\[\text{ We have } , \]
\[ \left( \cot\frac{x}{2} - \tan\frac{x}{2} \right)^2 \left( 1 - 2\text{ tan } x \cot2x \right)\]
\[\left( \cot^2 \frac{x}{2} - 2\cot\frac{x}{2}\tan\frac{x}{2} + \tan^2 \frac{x}{2} \right) \left\{ 1 - 2\text{ tan } x \left( \frac{\cot^2 x - 1}{2\text{ cot } x} \right) \right\}\]
\[\left( \cot^2 \frac{x}{2} - 2 + \tan^2 \frac{x}{2} \right)\left\{ 1 - \text{ tan } x \left( \frac{\cot^2 x - 1}{\text{ cot } x} \right) \right\}\]
\[\left( \cot^2 \frac{x}{2} + \tan^2 \frac{x}{2} - 2 \right)\left( 1 - \frac{\text{ cot } x - \text{ tan } x}{\text{ cot } x} \right)\]
\[\left( \cot^2 \frac{x}{2} + \tan^2 \frac{x}{2} - 2 \right)\left( \tan^2 x \right)\]
\[\left( \cot^2 \frac{x}{2} + \tan^2 \frac{x}{2} - 2 \right) \left( \frac{2\tan\frac{x}{2}}{1 - \tan^2 \frac{x}{2}} \right)^2\]
\[= \frac{1}{\left( 1 - \tan^2 \frac{x}{2} \right)^2}\left( 4 + 4 \tan^4 \frac{x}{2} - 8 \tan^2 \frac{x}{2} \right)\]
\[ = \frac{1}{\left( 1 - \tan^2 \frac{x}{2} \right)^2}\left( 4 - 8 \tan^2 \frac{x}{2} + 4 \tan^4 \frac{x}{2} \right)\]
\[ = \frac{4}{\left( 1 - \tan^2 \frac{x}{2} \right)^2} \left\{ \left( \tan^2 \frac{x}{2} \right)^2 - 2\left( \tan^2 \frac{x}{2} \right) + 1 \right\}\]
\[ = \frac{4 \left( \tan^2 \frac{x}{2} - 1 \right)^2}{\left( 1 - \tan^2 \frac{x}{2} \right)^2}\]
\[ = 4\]
APPEARS IN
RELATED QUESTIONS
Prove that: \[\frac{\sin 2x}{1 + \cos 2x} = \tan x\]
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: \[\left( \cos \alpha + \cos \beta^2 \right) + \left( \sin \alpha + \sin \beta \right)^2 = 4 \cos^2 \left( \frac{\alpha - \beta}{2} \right)\]
Prove that: \[\cos 4x = 1 - 8 \cos^2 x + 8 \cos^4 x\]
Show that: \[3 \left( \sin x - \cos x \right)^4 + 6 \left( \sin x + \cos \right)^2 + 4 \left( \sin^6 x + \cos^6 x \right) = 13\]
Prove that: \[\cos^6 A - \sin^6 A = \cos 2A\left( 1 - \frac{1}{4} \sin^2 2A \right)\]
Prove that: \[\cot^2 x - \tan^2 x = 4 \cot 2 x \text{ cosec } 2 x\]
Prove that \[\sin 3x + \sin 2x - \sin x = 4 \sin x \cos\frac{x}{2} \cos\frac{3x}{2}\]
If \[\tan A = \frac{1}{7}\] and \[\tan B = \frac{1}{3}\] , show that cos 2A = sin 4B
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
(ii) \[\tan\alpha \tan\beta = \frac{c - a}{c + a}\]
If \[\cos\alpha + \cos\beta = 0 = \sin\alpha + \sin\beta\] , then prove that \[\cos2\alpha + \cos2\beta = - 2\cos\left( \alpha + \beta \right)\] .
\[\tan x + \tan\left( \frac{\pi}{3} + x \right) - \tan\left( \frac{\pi}{3} - x \right) = 3 \tan 3x\]
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 \[\tan\frac{x}{2} = \frac{m}{n}\] , then write the value of m sin x + n cos x.
If \[\frac{\pi}{2} < x < \pi,\] the write the value of \[\sqrt{2 + \sqrt{2 + 2 \cos 2x}}\] in the simplest form.
If \[\frac{\pi}{2} < x < \pi\], then write the value of \[\frac{\sqrt{1 - \cos 2x}}{1 + \cos 2x}\] .
If \[\pi < x < \frac{3\pi}{2}\], then write the value of \[\sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}}\] .
The value of \[\tan x \sin \left( \frac{\pi}{2} + x \right) \cos \left( \frac{\pi}{2} - x \right)\]
If \[A = 2 \sin^2 x - \cos 2x\] , then A lies in the interval
If \[\tan \left( \pi/4 + x \right) + \tan \left( \pi/4 - x \right) = \lambda \sec 2x, \text{ then } \]
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 \[\cos \left( 36° - A \right) \cos \left( 36° + A \right) + \cos \left( 54° - A \right) \cos \left( 54° + A \right)\] 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 \[\tan\alpha = \frac{1}{7}, \tan\beta = \frac{1}{3}\], then
\[\cos2\alpha\] is equal to
The value of `cos^2 48^@ - sin^2 12^@` is ______.
If A = cos2θ + sin4θ for all values of θ, then prove that `3/4` ≤ A ≤ 1.
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 tanθ + sinθ = m and tanθ – sinθ = n, then prove that m2 – n2 = 4sinθ tanθ
[Hint: m + n = 2tanθ, m – n = 2sinθ, then use m2 – n2 = (m + n)(m – n)]
The value of cos248° – sin212° is ______.
[Hint: Use cos2A – sin2 B = cos(A + B) cos(A – B)]
