Trigonometric Ratios & Identities · Mathematics · MHT CET

The value of $\cos 20^{\circ}+2 \sin ^2 55^{\circ}-\sqrt{2} \sin 65^{\circ}$ is

The maximum value of $\left(\cos \alpha_1\right) \cdot\left(\cos \alpha_2\right) \ldots .\left(\cos \alpha_n\right)$ under the constraints $0 \leq \alpha_1, \alpha_2, \ldots ., \alpha_n \leq \frac{\pi}{2}$ and $\left(\cot \alpha_1\right) \cdot\left(\cot \alpha_2\right) \ldots\left(\cot \alpha_n\right)=1$ is

If $\mathrm{A}+\mathrm{B}=225^{\circ}$, then $\frac{\cot \mathrm{A}}{1+\cot \mathrm{A}} \cdot \frac{\cot \mathrm{B}}{1+\cot \mathrm{B}}$, if it exists, is equal to

The value of $\begin{aligned} \cos \left(18^{\circ}-\mathrm{A}\right) \cos \left(18^{\circ}+\mathrm{A}\right) -\cos \left(72^{\circ}-\mathrm{A}\right) \cos \left(72^{\circ}+\mathrm{A}\right) \text { is equal to }\end{aligned}$

$$ \cos ^3\left(\frac{\pi}{8}\right) \cos \left(\frac{3 \pi}{8}\right)+\sin ^3\left(\frac{\pi}{8}\right) \sin \left(\frac{3 \pi}{8}\right)=$$

If $\alpha+\beta=\frac{\pi}{2}$ and $\beta+\gamma=\alpha$, then $\tan \alpha$ equals

If $\mathrm{A}>\mathrm{B}$ and $\tan \mathrm{A}-\tan \mathrm{B}=x$ and $\cot \mathrm{B}-\cot \mathrm{A}=y$, then $\cot (\mathrm{A}-\mathrm{B})=$

The value of the expression $\sqrt{3} \operatorname{cosec} 20^{\circ}-\sec 20^{\circ}$ is equal to

If $\sin (\theta-\alpha), \sin \theta$ and $\sin (\theta+\alpha)$ are in H.P., then the value of $\cos ^2 \theta$ is

Let $\alpha$ and $\beta$ be two real roots of the equation $(k+1) \tan ^2 x-\sqrt{2} \lambda \tan x=(1-k)$ where $k(\neq-1)$ and $\lambda$ are real numbers. If $\tan ^2(\alpha+\beta)=50$, then a value of $\lambda$ is

If $\tan x=\frac{3}{4}$ and $\pi< x< \frac{3 \pi}{2}$, then $\cos \frac{x}{2}=$ ___________

The approximate value of $\cos \left(30^{\circ}, 30^{\prime}\right)$ is given that $1^{\circ}=0.0175^{\circ}$ and $\cos 30^{\circ}=0.8660$

If $\alpha+\beta+\gamma=\pi$, then the expression $\sin ^2 \alpha+\sin ^2 \beta-\sin ^2 \gamma$ has the value

If $$\mathrm{a} \cos 2 \theta+\mathrm{b} \sin 2 \theta=\mathrm{c}$$ has $$\alpha$$ and $$\beta$$ as its roots, then the value of $$\tan \alpha+\tan \beta$$ is

If $$\sin (\theta-\alpha), \sin \theta$$ and $$\sin (\theta+\alpha)$$ are in H.P., then the value of $$\cos 2 \theta$$ is

The value of $$\begin{aligned} \cos \left(18^{\circ}-\mathrm{A}\right) \cdot \cos ( & \left.18^{\circ}+\mathrm{A}\right) \\ & -\cos \left(72^{\circ}-\mathrm{A}\right) \cos \left(72^{\circ}+\mathrm{A}\right) \text { is }\end{aligned}$$

$$\cos ^2 48^{\circ}-\sin ^2 12^{\circ}=$$ _________, if $$\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}$$

If $$\tan \theta=\frac{\sin \alpha-\cos \alpha}{\sin \alpha+\cos \alpha}, 0 \leq \alpha \leq \frac{\pi}{2}$$, then the value of $$\cos 2 \theta$$ is

The value of $$\tan \left(\frac{\pi}{8}\right)$$ is _________.

The value of $$\tan \frac{\pi}{8}$$ is

If $$\cos 2 B=\frac{\cos (A+C)}{\cos (A-C)}$$. Then $$\tan A, \tan B, \tan C$$ are in

If $$\sin 18^{\circ}=\frac{\sqrt{5}-1}{4}$$, then $$\cos ^2 48^{\circ}-\sin ^2 12^{\circ}$$ has the value

If $$\cot (A+B)=0$$, then $$\sin (A+2 B)$$ is equal to

$$\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+4 \tan 4 \mathrm{~A}+8 \cot 8 \mathrm{~A}=$$

If $$a \sin \theta=b \cos \theta$$, where $$a, b \neq 0$$, then $$a\cos 2 \theta+b \sin 2 \theta=$$

If $$\sin (y+z-x), \sin (z+x-y)$$ and $$\sin (x+y-z)$$ are in AP, then

If $$2 \cos \theta=x+\frac{1}{x}$$, then $$2 \cos 3 \theta=$$

$$\tan 3 \mathrm{~A} \cdot \tan 2 \mathrm{~A} \cdot \tan \mathrm{A}=$$

If $$\frac{\cos (A+B)}{\cos (A-B)}=\frac{\sin (C+D)}{\sin (C-D)}$$, then $$\tan A \tan B \tan C=$$

If $$\cos x=\frac{24}{25}$$ and $$x$$ lięs in first quadrant, then $$\sin \frac{x}{2}+\cos \frac{x}{2}=$$

The value of $$\sin 18^{\circ}$$ is

$$\frac{1-\sin \theta+\cos \theta}{1-\sin \theta-\cos \theta}=$$

If $A$ and $B$ are supplementary angles, then $\sin ^2 \frac{A}{2}+\sin ^2 \frac{B}{2}=$

$$\begin{aligned} & \cos \left(36^{\circ}-A\right) \cos \left(36^{\circ}+A\right)+\cos \left(54^{\circ}+A\right) \cos \\ & \left(54^{\circ}-A\right)= \end{aligned}$$

If $$x+y=\frac{\pi}{2}$$, then the maximum value of $$\sin x \cdot \sin y$$ is

If $$a=\sin 175^{\circ}+\cos 175^{\circ}$$, then

The polar co-ordinates of the point whose cartesian co-ordinates are $$(-2,-2)$$, are given by

If $$x \cos \theta+y \sin \theta=5, x \sin \theta-y \cos \theta=3$$, then the value of $$x^2+y^2=$$

If $$\sin \theta=-\frac{12}{13}, \cos \phi=-\frac{4}{5}$$ and $$\theta, \phi$$ lie in the third quadrant, then $$\tan (\theta-\phi)=$$

$$\frac{1-2\left[\cos 60^{\circ}-\cos 80^{\circ}\right]}{2 \sin 10^{\circ}}=\ldots \ldots$$

The value of $\sin 18^{\circ}$ is $\qquad$

If $\theta=\frac{17 \pi}{3}$ then, $\tan \theta-\cot \theta=\ldots$