Trigonometric Ratios & Identities

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

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

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

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

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

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

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

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

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

\(\tan 3 \mathrm{~A} \cdot \tan 2 \mathrm{~A} \cdot \tan \mathrm{A}=\)

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

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

The value of \(\sin 18^{\circ}\) is

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

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

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

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

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

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

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

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

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

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

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

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