Trigonometric Ratios & Identities

\(\text { If } \frac{2 \sin \theta}{1+\cos \theta+\sin \theta}=y, \text { then } \frac{1-\cos \theta+\sin \theta}{1+\sin \theta}=\)

\(\frac{\sinh (x+y)+\sinh (x-y)}{\cosh (x+y)-\cosh (x-y)}=\)

If $\sinh x=\tan A$, then $|\tanh x|=$

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

If $A+B+C=\frac{\pi}{2}$, then $\sqrt{2} \cos \left(\frac{\pi}{4}-A\right)$
\(+\sqrt{2} \cos \left(\frac{\pi}{4}-B\right)+\sqrt{2} \cos \left(\frac{\pi}{4}-C\right)+1=\)

If $\sin (A+B) \sin (A-B)+\cos (A+B) \cos (A-B) =\frac{1}{2}$ and $0

If $\sin A=\frac{-7}{25}, \cos B=\frac{8}{17}, A$ does not lie in the 3rd quadrant and $B$ does not lie in the 1st quadrant, then $8 \tan A-5 \cot B=$

If $\sin \theta-\cos \theta=\frac{1}{\sqrt{3}}$, then $\sin (2 \theta)+\cos (4 \theta)+\sin (6 \theta)=$

If $a \tan \alpha+b \tan \beta=(a+b) \tan \left(\frac{\alpha+\beta}{2}\right)$ and $\alpha-\beta \neq 2 n \pi$ then $\frac{\cos \beta}{\cos \alpha}=$

If $\frac{5 \sinh 2 x}{7+6 \cosh 2 x}=\frac{3}{2}$, then $3 \tanh ^2 x+20 \tanh x=$

If $\sinh x=\frac{-1}{2}$, then $\tanh 2 x=$

If $\frac{1}{\sin 45^{\circ} \sin 46^{\circ}}+\frac{1}{\sin 46^{\circ} \sin 47^{\circ}}+\ldots$ up to 45 terms $=\frac{1}{\sin x^{\circ}}$, then $\sin \left(\frac{\pi}{2} x\right)=$

If $|\sin \alpha-\cos \alpha|=\frac{3}{4}$, then $|\sec 2 \alpha-\tan 2 \alpha|=$

\(\cos ^2 76^{\circ}+\sin ^2 46^{\circ}+\sin 76^{\circ} \cos 46^{\circ}=\)

Let $\alpha$ be the period of $3 \sin \frac{\pi x}{3}-\cos \frac{\pi x}{2}+\tan \frac{\pi x}{4}, \beta$ be the period of $\sin ^2\left(\frac{\pi}{7}+\frac{x}{4}\right)-\sin ^2\left(\frac{\pi}{7}-\frac{x}{4}\right)$, and $\gamma$ be the peri...

If $\theta$ does not lie in the second quadrant and $\tan \theta=\frac{-3}{4}$, then $\tan \frac{\theta}{2}+\sin 2 \theta=$

If $\tan A=\frac{2}{3}$, then $\sin 4 A=$

If $\theta=\frac{\pi}{12}$ and $x=\log \left(\cot \left(\frac{\pi}{4}+\theta\right)\right)$, then $\cosh x=$

If $A$ and $B(A>B)$ are acute angles, $\sin (A-B)=\frac{16}{65}$ and $\sin B=\frac{5}{13}$, then $\tan A+\cot A=$

$2 \cosh (x+y) \sinh (x-y)+\sinh 2 y=$

\(\frac{\sqrt{2} \cos 45^{\circ}+\cos 56^{\circ}+\cos 58^{\circ}-\cos 66^{\circ}}{\sqrt{2} \cos 28^{\circ} \cos 29^{\circ} \sin 33^{\circ}}\)

If $\cos x+\cos y=p, \sin x+\sin y=q$, then $\cos \left(\frac{x-y}{2}\right)=$

If $\tanh x=\frac{1}{2}$, then $\sinh 2 x-\operatorname{sech} 2 x=$

If $A+B+C=\frac{3 \pi}{2}$, then $4 \sin A \sin B \sin C+\cos 2 A+\cos 2 B+\cos 2 C=$

\(\frac{e^{4 x}+e^{-4 x}+14}{4\left(e^x-e^{-x}\right)^2}=\)

The ratio of the maximum and minimum values attained by the function $f(x)=1+2 \sin x+3 \cos ^2 x, 0 \leq x \leq \frac{2 \pi}{3}$ is

$\theta$ and $\alpha$ lie in $Q_3$. If $\cos (\theta-\alpha), \cos \theta, \cos (\theta+\alpha)$ are in harmonic progression, then $\cos \theta \sec \frac{\alpha}{2}=$

For $n \in \mathbf{N}$, if $f(n)=(\cos n x)(\sec x)^n$ and $g(n)=(\sin n x)(\sec x)^n$, then $f(2020)-f(2019)+(\tan x) g(2019)=$

If $\sin \theta \cosh \alpha=\tan x, \cos \theta \sinh \alpha=\sec x$, then $\cos 2 \theta \cosh 2 \alpha=$

If $\cot \left(\frac{A}{2}\right)=\sqrt{\frac{1+a}{1-a}} \cdot \cot \left(\frac{\theta}{2}\right)$, then $\cos \theta=$