Prove that \(\dfrac{\tan A + \sec A - 1}{\tan A - \sec A + 1} = \dfrac{1 + \sin A}{\cos A}\).
If \(\dfrac{2\sin\alpha}{1 + \cos\alpha + \sin\alpha} = y\), then prove that \(\dfrac{1 - \cos\alpha + \sin\alpha}{1 + \sin\alpha}\) is also equal to \(y\).
If \(m\sin\theta = n\sin(\theta + 2\alpha)\), then prove that \(\tan(\theta + \alpha)\cot\alpha = \dfrac{m + n}{m - n}\).
If \(\cos(\alpha + \beta) = \dfrac{4}{5}\) and \(\sin(\alpha - \beta) = \dfrac{5}{13}\), where \(\alpha\) lies between \(0\) and \(\dfrac{\pi}{4}\), find the value of \(\tan 2\alpha\).
If \(\tan x = \dfrac{b}{a}\), then find the value of \(\sqrt{\dfrac{a + b}{a - b}} + \sqrt{\dfrac{a - b}{a + b}}\).
Prove that \(\cos\theta\cos\dfrac{\theta}{2} - \cos3\theta\cos\dfrac{9\theta}{2} = \sin7\theta\sin8\theta\).
If \(a\cos\theta + b\sin\theta = m\) and \(a\sin\theta - b\cos\theta = n\), then show that \(a^2 + b^2 = m^2 + n^2\).
Find the value of \(\tan 22^{\circ}30'\).
Prove that \(\sin 4A = 4\sin A\cos^{3}A - 4\cos A\sin^{3}A\).
If \(\tan\theta + \sin\theta = m\) and \(\tan\theta - \sin\theta = n\), then prove that \(m^2 - n^2 = 4\sin\theta\tan\theta\).
If \(\tan(A + B) = p\) and \(\tan(A - B) = q\), then show that \(\tan 2A = \dfrac{p + q}{1 - pq}\).
If \(\cos\alpha + \cos\beta = 0 = \sin\alpha + \sin\beta\), then prove that \(\cos 2\alpha + \cos 2\beta = -2\cos(\alpha + \beta)\).
If \(\dfrac{\sin(x + y)}{\sin(x - y)} = \dfrac{a + b}{a - b}\), then show that \(\dfrac{\tan x}{\tan y} = \dfrac{a}{b}\).
If \(\tan\theta = \dfrac{\sin\alpha - \cos\alpha}{\sin\alpha + \cos\alpha}\), then show that \(\sin\alpha + \cos\alpha = \sqrt{2}\cos\theta\).
If \(\sin\theta + \cos\theta = 1\), then find the general value of \(\theta\).
Find the most general value of \(\theta\) satisfying the equations \(\tan\theta = -1\) and \(\cos\theta = \dfrac{1}{\sqrt{2}}\).
If \(\cot\theta + \tan\theta = 2\csc\theta\), then find the general value of \(\theta\).
If \(2\sin^{2}\theta = 3\cos\theta\), where \(0 \le \theta \le 2\pi\), then find the value(s) of \(\theta\).
If \(\sec x \cos 5x + 1 = 0\), where \(0 < x \le \dfrac{\pi}{2}\), then find the value of \(x\).
If \(\sin(\theta + \alpha) = a\) and \(\sin(\theta + \beta) = b\), then prove that \(\cos 2(\alpha - \beta) - 4ab \cos(\alpha - \beta) = 1 - 2a^2 - 2b^2\).
If \(\cos(\theta + \phi) = m \cos(\theta - \phi)\), then prove that \(\tan \theta = \dfrac{1 - m}{1 + m} \cot \phi\).
Find the value of the expression
\(3\left[\sin^4\left(\dfrac{3\pi}{2} - \alpha\right) + \sin^4(3\pi + \alpha)\right] - 2\left[\sin^6\left(\dfrac{\pi}{2} + \alpha\right) + \sin^6(5\pi - \alpha)\right].\)
1
If an equation \(a \cos 2\theta + b \sin 2\theta = c\) has \(\alpha\) and \(\beta\) as its roots, then prove that
\(\tan \alpha + \tan \beta = \dfrac{2b}{a + c}.\)
If \(x = \sec \phi - \tan \phi\) and \(y = \csc \phi + \cot \phi\), then show that \(xy + x - y + 1 = 0\).
If \(\theta\) lies in the first quadrant and \(\cos \theta = \dfrac{8}{17}\), then find the value of
\(\cos(30^{\circ} + \theta) + \cos(45^{\circ} - \theta) + \cos(120^{\circ} - \theta).\)
\(\dfrac{23}{17}\left(\dfrac{\sqrt{3} - 1}{2} + \dfrac{1}{\sqrt{2}}\right)\)
Find the value of the expression
\(\cos^4\dfrac{\pi}{8} + \cos^4\dfrac{3\pi}{8} + \cos^4\dfrac{5\pi}{8} + \cos^4\dfrac{7\pi}{8}.\)
\(\dfrac{3}{2}\)
Find the general solution of the equation \(5\cos^2\theta + 7\sin^2\theta - 6 = 0\).
\(n\pi \pm \dfrac{\pi}{4}\)
Find the general solution of the equation
\(\sin x - 3 \sin 2x + \sin 3x = \cos x - 3 \cos 2x + \cos 3x.\)
\(\dfrac{n\pi}{2} \pm \dfrac{\pi}{8}\)
Find the general solution of the equation
\((\sqrt{3} - 1)\cos \theta + (\sqrt{3} + 1)\sin \theta = 2.\)
\(\theta = 2n\pi \pm \dfrac{\pi}{4} + \dfrac{\pi}{12}\)
If \(\sin\theta + \csc\theta = 2\), then \(\sin^2\theta + \csc^2\theta\) is equal to
1
4
2
None of these
If \(f(x)=\cos^2 x + \sec^2 x\), then
\(f(x) < 1\)
\(f(x)=1\)
\(2 < f(x) < 1\)
\(f(x) \ge 2\)
If \(\tan\theta=\dfrac{1}{2}\) and \(\tan\phi=\dfrac{1}{3}\), then the value of \(\theta+\phi\) is
\(\dfrac{\pi}{6}\)
\(\pi\)
0
\(\dfrac{\pi}{4}\)
Which of the following is not correct?
\(\sin\theta = -\dfrac{1}{5}\)
\(\cos\theta = 1\)
\(\sec\theta = \dfrac{1}{2}\)
\(\tan\theta = 20\)
The value of \(\tan1^\circ\tan2^\circ\tan3^\circ\dots\tan89^\circ\) is
0
1
\(\dfrac{1}{2}\)
Not defined
The value of \(\dfrac{1-\tan^2 15^\circ}{1+\tan^2 15^\circ}\) is
1
\(\sqrt{3}\)
\(\dfrac{\sqrt{3}}{2}\)
2
The value of \(\cos1^\circ\cos2^\circ\cos3^\circ\dots\cos179^\circ\) is
\(\dfrac{1}{\sqrt{2}}\)
0
1
-1
If \(\tan\theta=3\) and \(\theta\) lies in the third quadrant, then the value of \(\sin\theta\) is
\(\dfrac{1}{\sqrt{10}}\)
\(-\dfrac{1}{\sqrt{10}}\)
\(-\dfrac{3}{\sqrt{10}}\)
\(\dfrac{3}{\sqrt{10}}\)
The value of \(\tan75^\circ - \cot75^\circ\) is equal to
\(2\sqrt{3}\)
2 + \(\sqrt{3}\)
2 - \(\sqrt{3}\)
1
Which of the following is correct?
\(\sin1^\circ > \sin1\)
\(\sin1^\circ < \sin1\)
\(\sin1^\circ = \sin1\)
\(\sin1^\circ = \dfrac{\pi}{18}\sin1\)
If \(\tan\alpha=\dfrac{m}{m+1}\), \(\tan\beta=\dfrac{1}{2m+1}\), then \(\alpha+\beta\) is equal to
\(\dfrac{\pi}{2}\)
\(\dfrac{\pi}{3}\)
\(\dfrac{\pi}{6}\)
\(\dfrac{\pi}{4}\)
The minimum value of \(3\cos x + 4\sin x + 8\) is
5
9
7
3
The value of \(\tan3A - \tan2A - \tan A\) is equal to
\(\tan3A\tan2A\tan A\)
\(-\tan3A\tan2A\tan A\)
\(\tan A\tan2A - \tan2A\tan3A - \tan3A\tan A\)
None of these
The value of \(\sin(45^\circ + \theta) - \cos(45^\circ - \theta)\) is
\(2\cos\theta\)
\(2\sin\theta\)
1
0
The value of \(\cot\left(\dfrac{\pi}{4}+\theta\right)\cot\left(\dfrac{\pi}{4}-\theta\right)\) is
-1
0
1
Not defined
\(\cos2\theta\cos2\phi + \sin^2(\theta-\phi) - \sin^2(\theta+\phi)\) is equal to
\(\sin2(\theta+\phi)\)
\(\cos2(\theta+\phi)\)
\(\sin2(\theta-\phi)\)
\(\cos2(\theta-\phi)\)
The value of \(\cos12^\circ + \cos84^\circ + \cos156^\circ + \cos132^\circ\) is
\(\dfrac{1}{2}\)
1
\(-\dfrac{1}{2}\)
\(\dfrac{1}{8}\)
If \(\tan A=\dfrac{1}{2}\), \(\tan B=\dfrac{1}{3}\), then \(\tan(2A+B)\) is equal to
1
2
3
4
The value of \(\sin\dfrac{\pi}{10} - \sin\dfrac{13\pi}{10}\) is
\(\dfrac{1}{2}\)
\(-\dfrac{1}{2}\)
\(-\dfrac{1}{4}\)
1
The value of \(\sin50^\circ - \sin70^\circ + \sin10^\circ\) is equal to
1
0
\(\dfrac{1}{2}\)
2
If \(\sin\theta + \cos\theta = 1\), then the value of \(\sin2\theta\) is equal to
1
\(\dfrac{1}{2}\)
0
-1
If \(\alpha + \beta = \dfrac{\pi}{4}\), then the value of \((1+\tan\alpha)(1+\tan\beta)\) is
1
2
-2
Not defined
If \(\sin\theta = -\dfrac{4}{5}\) and \(\theta\) lies in the third quadrant, then the value of \(\cos\dfrac{\theta}{2}\) is
\(\dfrac{1}{5}\)
-\(\dfrac{1}{\sqrt{10}}\)
-\(\dfrac{1}{\sqrt{5}}\)
\(\dfrac{1}{\sqrt{10}}\)
Number of solutions of the equation \(\tan x + \sec x = 2\cos x\) lying in the interval \([0,2\pi]\) is
0
1
2
3
The value of \(\sin\dfrac{\pi}{18} + \sin\dfrac{\pi}{9} + \sin\dfrac{2\pi}{9} + \sin\dfrac{5\pi}{18}\) is given by
\(\sin\dfrac{7\pi}{18} + \sin\dfrac{4\pi}{9}\)
1
\(\cos\dfrac{\pi}{6} + \cos\dfrac{3\pi}{7}\)
\(\cos\dfrac{\pi}{9} + \sin\dfrac{\pi}{9}\)
If \(A\) lies in the second quadrant and \(3\tan A + 4 = 0\), then the value of \(2\cot A - 5\cos A + \sin A\) is equal to
-\(\dfrac{53}{10}\)
\(\dfrac{23}{10}\)
\(\dfrac{37}{10}\)
\(\dfrac{7}{10}\)
The value of \(\cos^2 48^\circ - \sin^2 12^\circ\) is
\(\dfrac{\sqrt{5}+1}{8}\)
\(\dfrac{\sqrt{5}-1}{8}\)
\(\dfrac{\sqrt{5}+1}{5}\)
\(\dfrac{\sqrt{5}+1}{2\sqrt{2}}\)
If \(\tan\alpha=\dfrac{1}{7}\), \(\tan\beta=\dfrac{1}{3}\), then \(\cos2\alpha\) is equal to
\(\sin2\beta\)
\(\sin4\beta\)
\(\sin3\beta\)
\(\cos2\beta\)
If \(\tan\theta=\dfrac{a}{b}\), then \(b\cos2\theta + a\sin2\theta\) is equal to
a
b
\(\dfrac{a}{b}\)
None
If for real values of \(x\), \(\cos\theta = x + \dfrac{1}{x}\), then
\(\theta\) is an acute angle
\(\theta\) is right angle
\(\theta\) is an obtuse angle
No value of \(\theta\) is possible
The value of \( \dfrac{\sin 50^\circ}{\sin 130^\circ} \) is ____.
1
If \( k = \sin\left(\dfrac{\pi}{18}\right) \sin\left(\dfrac{5\pi}{18}\right) \sin\left(\dfrac{7\pi}{18}\right) \), then the numerical value of \( k \) is ____.
\(\dfrac{1}{8}\)
If \( \tan A = \dfrac{1 - \cos B}{\sin B} \), then \( \tan 2A = ____ \).
\(\tan \beta\)
If \(\sin x + \cos x = a\), then
(i) \( \sin^6 x + \cos^6 x = ____ \)
(ii) \(|\sin x - \cos x| = ____ \)
\(\dfrac{1}{4}[4 - 3(a^2 - 1)^2]\), \(\sqrt{2 - a^2}\)
In a triangle ABC with \(\angle C = 90^\circ\) the equation whose roots are \(\tan A\) and \(\tan B\) is ____.
\(x^2 - \dfrac{2}{\sin 2A} x + 1\)
\(3(\sin x - \cos x)^4 + 6(\sin x + \cos x)^2 + 4(\sin^6 x + \cos^6 x) = ____\).
13
Given \(x > 0\), the values of \( f(x) = -3 \cos \sqrt{3 + x + x^2} \) lie in the interval ____.
[ -3 , 3 ]
The maximum distance of a point on the graph of the function \( y = \sqrt{3} \sin x + \cos x \) from x-axis is ____.
2
If \( \, \tan A = \dfrac{1 - \cos B}{\sin B} \, \) then \( \, \tan 2A = \tan B \, \).
True
The equality \( \, \sin A + \sin 2A + \sin 3A = 3 \, \) holds for some real value of \(A\).
False
\( \, \sin 10^\circ \, \) is greater than \( \, \cos 10^\circ \, \).
False
\( \cos \dfrac{2\pi}{15} \, \cos \dfrac{4\pi}{15} \, \cos \dfrac{8\pi}{15} \, \cos \dfrac{16\pi}{15} = \dfrac{1}{16} \).
True
One value of \(\theta\) which satisfies the equation \( \, \sin^4 \theta - 2 \sin^2 \theta - 1 \, \) lies between \(0\) and \(2\pi\).
False
If \( \, \csc x = 1 + \cot x \, \) then \( x = 2n\pi \) or \( x = 2n\pi + \dfrac{\pi}{2} \).
True
If \( \, \tan \theta + \tan 2\theta + \sqrt{3} \, \tan \theta \, \tan 2\theta = \sqrt{3} \, \), then \( \, \theta = \dfrac{n\pi}{3} + \dfrac{\pi}{9} \, \).
True
If \( \, \tan(\pi \cos \theta) = \cot(\pi \sin \theta) \, \) then \( \, \theta - \dfrac{\pi}{4} = \pm \dfrac{1}{2\sqrt{2}} \, \).
True
Match the items in Column A with their corresponding expressions in Column B using the table below.
| Column A | Column B |
|---|---|
(a) \(\sin(x+y)\sin(x-y)\) | (i) \(\cos^2 x - \sin^2 y\) |
(b) \(\cos(x+y)\cos(x-y)\) | (ii) \(\dfrac{1-\tan\theta}{1+\tan\theta}\) |
(c) \(\cot\left(\dfrac{\pi}{4} + \theta\right)\) | (iii) \(\dfrac{1+\tan\theta}{1-\tan\theta}\) |
(d) \(\tan\left(\dfrac{\pi}{4} + \theta\right)\) | (iv) \(\sin^2 x - \sin^2 y\) |
| Column A | Matched Item from Column B |
|---|---|
(a) \(\sin(x+y)\sin(x-y)\) | (iv) \(\sin^2 x - \sin^2 y\) |
(b) \(\cos(x+y)\cos(x-y)\) | (i) \(\cos^2 x - \sin^2 y\) |
(c) \(\cot\left(\dfrac{\pi}{4} + \theta\right)\) | (ii) \(\dfrac{1-\tan\theta}{1+\tan\theta}\) |
(d) \(\tan\left(\dfrac{\pi}{4} + \theta\right)\) | (iii) \(\dfrac{1+\tan\theta}{1-\tan\theta}\) |