Trigonometric Ratios & Identities

If $\sec \theta+\tan \theta=\frac{1}{3}$, then the quadrant in which $2 \theta$ lies is

If $(\alpha+\beta)$ is not a multiple of $\frac{\pi}{2}$ and $3 \sin (\alpha-\beta)=5 \cos (\alpha+\beta)$, then $\tan \left(\frac{\pi}{4}+\alpha\right)+4 \tan \left(\frac{\pi}{4}+\beta\right)=$

The value of $5 \cos \theta+3 \cos \left(\theta+\frac{\pi}{3}\right)+3$ lies between

Statement $(\mathrm{S} 1) \sin 55^{\circ}+\sin 53^{\circ}-\sin 19^{\circ}-\sin 17^{\circ}=\cos 2^{\circ}$
Statement (S2) Range of $\frac{1}{3-\cos 2 x}$ is $\left[\frac{1}{4}, \frac{1}{2}\right]$
Which one of the following is correct?

By considering $1^{\prime}=0.0175$, he approximate value of $\cot 45^{\circ} 2^{\prime}$ is

$\sin^2 16^\circ - \sin^2 76^\circ = $

The maximum value of $12\sin x - 5\cos x + 3$ is

$ \tan 6^\circ + \tan 42^\circ + \tan 66^\circ + \tan 78^\circ = $

If $\sec (\theta+\alpha), \sec \theta$ and $\sec (\theta-\alpha)$ are in arithmetic progression, then $\sin ^2 \theta=$

If $\alpha=\log _e(2+\sqrt{3})$, then $\frac{\cosh \alpha}{1-\tanh \alpha}+\frac{\sinh \alpha}{1-\operatorname{coth} \alpha}=$

If $\cos \alpha+\cos \beta=a$ and $\sin \alpha+\sin \beta=b$, then match the items given in List-A with those of their values in List-B
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$\sin 21^{\circ} \cos 9^{\circ}-\cos 84^{\circ} \cos 6^{\circ}=$

If $(1+\sqrt{1+a}) \sin \alpha=(1+\sqrt{1-a}) \cos \alpha$ and $0

If $A=\frac{\pi}{24}$, then $\frac{\cos A+\cos 3 A+\cos 5 A+\cos 7 A}{\sin A+\sin 3 A+\sin 5 A+\sin 7 A}=$

$$ \begin{aligned} & \text { If }\left[1-\cos \left(\frac{\pi}{2}+\alpha\right)+\sin \left(\frac{3 \pi}{2}+\alpha\right)\right]^2 +\left[1-\sin \left(\frac{3 \pi}{2}-\alpha\right)-\cos \left(\frac{3 \pi}{2}+\alpha\right)\right]^2 \\ & =a+b ...

\(\frac{1}{\cos 290^{\circ}}+\frac{1}{\sqrt{3} \sin 250^{\circ}}=\)

If $\tan A+\tan B=x$ and $\cot A+\cot B=y$, then $\tan (A+B)=$

The range of $\frac{1}{\sin ^2 x+3 \sin x \cos x+5 \cos ^2 x}$ is

If $\cos h x=\frac{5}{4}$, then $\tan h 3 x=$

If $\cot x \cot y=a$ and $x+y=\frac{\pi}{6}$, then the quadratic equation satisfying $\cot x$ and $\cot y$ is

\(\sin ^2 \frac{2 \pi}{3}+\cos ^2 \frac{5 \pi}{6}-\tan ^2 \frac{3 \pi}{4}=\)

Which of the following trigonometric values are negative?
I. \(\sin \left(-292^{\circ}\right)\)
II. \(\tan \left(-190^{\circ}\right)\)
III. \(\cos \left(-207^{\circ}\right)\)
IV. \(\cot \left(-222^{\circ}\right)\)

$$\begin{aligned} & \frac{1}{\sin 1^{\circ} \sin 2^{\circ}}+\frac{1}{\sin 2^{\circ} \sin 3^{\circ}}+\ldots +\frac{1}{\sin 89^{\circ}+\sin 90^{\circ}}= \end{aligned}$$

If \(\sin \theta=-\frac{3}{4}\), then \(\sin 2 \theta=\)

If \(\sin ^4 \theta \cos ^2 \theta=\sum_\limits{n=0}^{\infty} a_{2 n} \cos 2 n \theta\), then the least \(n\) for which \(a_{2 n}=0\) is

If \(\operatorname{cosech} x=\frac{4}{5}\), then \(\sinh x=\)

\(\frac{\cos \theta}{1-\tan \theta}+\frac{\sin \theta}{1-\cot \theta}=\)

The value of \(\frac{\sin \theta+\sin 3 \theta}{\cos \theta+\cos 3 \theta}\) is

If the identity \(\cos ^4 \theta=a \cos 4 \theta+b \cos 2 \theta+c\) holds for some \(a, b, c \in Q\) then \((a, b, c)=\)

If \((1+\tan 1^{\circ})(1+\tan 2^{\circ}) \ldots(1+\tan 45^{\circ})=2^n,\) then \(n=\)

\(1+\sec ^2 x \sin ^2 x=\)

The value of \(\tan \left(\frac{7 \pi}{8}\right)\) is

A true statement among the following identities is

If \(A+B+C=\pi, \cos B=\cos A \cos C\), then \(\tan A \tan C=\)

The sides of a triangle inscribed in a given circle subtend angles \(\alpha, \beta, \gamma\) at the center. The minimum value of the AM of \(\cos \left(\alpha+\frac{\pi}{2}\right), \cos \left(\beta+\frac{\pi}{2}\right)\) and $$\cos \left(\g...

If \(\tan \beta=\frac{\tan \alpha+\tan \gamma}{1+\tan \alpha \tan \gamma}\), then \(\frac{\sin 2 \alpha+\sin 2 \gamma}{1+\sin 2 \alpha \sin 2 \gamma}\) is equal to

If \(\cos \theta=-\sqrt{\frac{3}{2}}\) and \(\sin \alpha=\frac{-3}{5}\), where '\(\theta\)' does not lie in the third quadrant, then the value of \(\frac{2 \tan \alpha+\sqrt{3} \tan \theta}{\cot ^2 \theta+\cos \alpha}\) is equal to

What is the value of \(\cos \left(22 \frac{1}{2}\right)^{\circ}\) ?

\(\sinh (x+y) \cosh (x-y)\) is equal to

If \(A+B+C=\frac{3 \pi}{2}\), then \(\cos 2 A+\cos 2 B+\cos 2 C\) is equal to

If \(\cos \frac{\pi}{4} \cos \frac{\pi}{8} \cos \frac{\pi}{16} \cos \frac{\pi}{32}=2^m \operatorname{cosec} \frac{\pi}{n}\), then \(m+n\) is equal to

Let \(\theta\) be an angle in the standard position such that the point \((-5,12)\) lies on its terminal side, then

If \(\sinh u=\tan \theta\), then \(\cosh u\) is equal to

If \(\sin \alpha - \cos \alpha = m\) and \(\sin 2\alpha = n - {m^2}\), where \(- \sqrt 2 \le m \le \sqrt 2\), then n is equal to

\(\tan 2 \alpha \cdot \tan (30 Y-\alpha)+\tan 2 \alpha \cdot \tan (60 Y-\alpha)+\tan (60 \Upsilon-\alpha) \cdot \tan (30 \gamma-\alpha)\) is equal to

The maximum value of \(f(x)=\sin (x)\) in the interval \(\left[\frac{-\pi}{2}, \frac{\pi}{2}\right]\) is

When the coordinate axes are rotated through an angle 135\(\Upsilon\), the coordinates of a point \(P\) in the new system are known to be \((4,-3)\). Then find the coordinates of \(P\) in the original system.

Mean of the values \(\sin ^2 10 Y, \sin ^2 20 Y, \sin ^2 30 Y, \ldots \ldots \ldots ., \sin ^2 90 Y\) is

In \(\triangle A B C \cdot \frac{a+b+c}{B C+A B}+\frac{a+b+c}{A C+A B}=3\), then \(\tan \frac{C}{8}\) is equal to

If \(f(x)=\frac{\cot x}{1+\cot x}\) and \(\alpha+\beta=\frac{5 \pi}{4}\), then the value of \(f(\alpha) f(\beta)\) is equal to