Advertisements
Advertisements
Question
The value of \[\cos^2 \left( \frac{\pi}{6} + x \right) - \sin^2 \left( \frac{\pi}{6} - x \right)\] is
Options
\[\frac{1}{2} \cos 2x\]
0
\[- \frac{1}{2} \cos 2x\]
\[\frac{1}{2}\]
Advertisements
Solution
\[\frac{1}{2} \cos 2x\]
\[\text{ We have, } \]
\[ \cos^2 \left( \frac{\pi}{6} + x \right) - \sin^2 \left( \frac{\pi}{6} - x \right)\]
\[ = \cos^2 \left( \frac{\pi}{6} + x \right) - \cos^2 \left[ \frac{\pi}{2} - \left( \frac{\pi}{6} - x \right) \right]\]
\[ = \cos^2 \left( \frac{\pi}{6} + x \right) - \cos^2 \left( \frac{\pi}{3} + x \right)\]
\[ = \left[ \cos\left( \frac{\pi}{6} + x \right) + \cos\left( \frac{\pi}{3} + x \right) \right]\left[ \cos\left( \frac{\pi}{6} + x \right) - \cos\left( \frac{\pi}{3} + x \right) \right]\]
\[ = 2\cos\left( \frac{\frac{\pi}{6} + x + \frac{\pi}{3} + x}{2} \right) \cos\left( \frac{\frac{\pi}{6} + x - \frac{\pi}{3} - x}{2} \right) 2\sin\left( \frac{\frac{\pi}{6} + x + \frac{\pi}{3} + x}{2} \right) \sin\left( \frac{\frac{\pi}{3} + x - \frac{\pi}{6} - x}{2} \right)\]
\[ = 4\cos\left( \frac{\pi}{4} + x \right)\cos\left( - \frac{\pi}{12} \right) \sin\left( \frac{\pi}{4} + x \right) \sin\left( \frac{\pi}{12} \right)\]
\[ = 4\cos\left( \frac{\pi}{4} + x \right)\cos\left( \frac{\pi}{12} \right) \sin\left( \frac{\pi}{4} + x \right) \sin\left( \frac{\pi}{12} \right)\]
\[ = \left[ 2\sin\left( \frac{\pi}{4} + x \right)\cos\left( \frac{\pi}{4} + x \right) \right]\left[ 2 \sin\left( \frac{\pi}{12} \right)\cos\left( \frac{\pi}{12} \right) \right]\]
\[ = \sin\left( \frac{\pi}{2} + 2x \right)\sin\frac{\pi}{6}\]
\[ = \cos2x \times \frac{1}{2}\]
\[ = \frac{1}{2}\cos2x\]
APPEARS IN
RELATED QUESTIONS
Prove that: \[\sqrt{\frac{1 - \cos 2x}{1 + \cos 2x}} = \tan x\]
Prove that: \[\frac{\sin 2x}{1 + \cos 2x} = \tan x\]
Prove that: \[\sqrt{2 + \sqrt{2 + 2 \cos 4x}} = 2 \text{ cos } x\]
Prove that: \[\frac{\cos 2 x}{1 + \sin 2 x} = \tan \left( \frac{\pi}{4} - x \right)\]
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: \[\sin 4x = 4 \sin x \cos^3 x - 4 \cos x \sin^3 x\]
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: \[\cot^2 x - \tan^2 x = 4 \cot 2 x \text{ cosec } 2 x\]
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 0 ≤ x ≤ π and x lies in the IInd quadrant such that \[\sin x = \frac{1}{4}\]. Find the values of \[\cos\frac{x}{2}, \sin\frac{x}{2} \text{ and } \tan\frac{x}{2}\]
If \[\tan A = \frac{1}{7}\] and \[\tan B = \frac{1}{3}\] , show that cos 2A = sin 4B
If \[2 \tan \alpha = 3 \tan \beta,\] prove that \[\tan \left( \alpha - \beta \right) = \frac{\sin 2\beta}{5 - \cos 2\beta}\] .
If \[\cos \alpha + \cos \beta = \frac{1}{3}\] and sin \[\sin\alpha + \sin \beta = \frac{1}{4}\] , prove that \[\cos\frac{\alpha - \beta}{2} = \pm \frac{5}{24}\]
Prove that: \[\sin 5x = 5 \sin x - 20 \sin^3 x + 16 \sin^5 x\]
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: \[\cos\frac{\pi}{15} \cos \frac{2\pi}{15} \cos \frac{3\pi}{15} \cos \frac{4\pi}{15} \cos \frac{5\pi}{15} \cos\frac{6\pi}{15} \cos \frac{7\pi}{15} = \frac{1}{128}\]
If \[\frac{\pi}{2} < x < \pi\], then write the value of \[\frac{\sqrt{1 - \cos 2x}}{1 + \cos 2x}\] .
If \[\frac{\pi}{4} < x < \frac{\pi}{2}\], then write the value of \[\sqrt{1 - \sin 2x}\] .
If \[\text{ sin } x + \text{ cos } x = a\], find the value of \[\left|\text { sin } x - \text{ cos } x \right|\] .
The value of \[\cos \frac{\pi}{65} \cos \frac{2\pi}{65} \cos \frac{4\pi}{65} \cos \frac{8\pi}{65} \cos \frac{16\pi}{65} \cos \frac{32\pi}{65}\] is
If in a \[∆ ABC, \tan A + \tan B + \tan C = 0\], then
If \[\sin \alpha + \sin \beta = a \text{ and } \cos \alpha - \cos \beta = b \text{ then } \tan \frac{\alpha - \beta}{2} =\]
If \[5 \sin \alpha = 3 \sin \left( \alpha + 2 \beta \right) \neq 0\] , then \[\tan \left( \alpha + \beta \right)\] is equal to
The value of \[\frac{\cos 3x}{2 \cos 2x - 1}\] is equal to
If \[\tan \left( \pi/4 + x \right) + \tan \left( \pi/4 - x \right) = \lambda \sec 2x, \text{ then } \]
If \[\tan x = t\] then \[\tan 2x + \sec 2x =\]
The value of \[\frac{\sin 5 \alpha - \sin 3\alpha}{\cos 5 \alpha + 2 \cos 4\alpha + \cos 3\alpha} =\]
The value of `cos pi/5 cos (2pi)/5 cos (4pi)/5 cos (8pi)/5` is ______.
The value of sin50° – sin70° + sin10° is equal to ______.
If sinθ = `(-4)/5` and θ lies in the third quadrant then the value of `cos theta/2` is ______.
