Trigonometric Equations, Functions, Identities and Ratios

The expression $\sqrt{2+\sqrt{2+\sqrt{2+2 \cos \left(8\theta \right)}}}$ is equal to what?

If $x=a\cos (90°-\theta )$ and $y=b\cot (90°-\theta )$ , then the value of is $\left(\frac{a^2}{x^2}-\frac{b^2}{y^2}\right)$. 

If $\left(\mathrm{cosec} \mathrm{\theta }-\sin \mathrm{\theta }\right)={\mathrm{a}}^3 \& \left(\sec \mathrm{\theta }-\cos \mathrm{\theta }\right)={\mathrm{b}}^3$ then, the value of ${\mathrm{a}}^2{\mathrm{b}}^2\left({\mathrm{a}}^2+{\mathrm{b}}^2\right)$ is _______.

Maximum value of $(2\sin \theta +3\cos \theta )$ is _____.

If $\tan \theta =\frac{A}{B}$, What should come in the place of ($?$) for the expression given below?

$\frac{A\sin \theta -B\cos \theta }{A\sin \theta +B\cos \theta }=?$

What is the maximum value of $\sin 3i\times \cos 2i+\cos 3i\times \sin 2i$?

If $2\cos \left(\theta \right)-\sin \left(\theta \right)=\frac{1}{\sqrt{2}}$ and $\left(0°<\theta <90°\right)$, then the value of $2\sin \left(\theta \right)+\cos \left(\theta \right)$ is:

If $\tan (9°)=\frac{p}{q}$, then find the value of $\frac{{\sec }^2\left(81°\right)}{\left(1+{\cot }^2\left(81°\right)\right)}$.

If $\mathrm{tan\theta }=\frac{\mathrm{x} \sin \mathrm{a}}{1-\mathrm{x} \cos \mathrm{a}}$  and  $\tan \mathrm{a}=\frac{\mathrm{y} \mathrm{sin\theta }}{1-\mathrm{y} \mathrm{cos\theta }}$, then $\frac{\mathrm{x}}{\mathrm{y}}$ equals to ______.

An aeroplane when 900 m high passes vertically above another aeroplane at an instant when their angles of elevation at same observing point are 60° and 45° respectively. Approximately, how many meters higher is the one than the other?

The angle of elevation of an aeroplane from a point on the ground is $45°$, after $15 \sec$ of flight, the elevation changes to $30°$, if the aeroplane is flying at a height of $3000 \mathrm{m}$, find the speed of the plane.

What is the value of $\left(1+\cot \mathrm{x}-\mathrm{cosec} \mathrm{x}\right)\left(1+\tan \mathrm{x}+\sec \right)$?

If $\mathrm{tanA}=\frac{1-\mathrm{cosB}}{\mathrm{sinB}}$, then what is$\frac{2\mathrm{tanA}}{1-{\tan }^2\mathrm{A}}$  equal to ______.

If $\sin A \cos A=\frac{1}{2}$, then find the value of $(\sin A -\cos A)$.

A kite is flying at a height of $75\mathrm{m}$ above the ground. The inclination of the string with the ground is ${60}^0$ . Find the length of the string, assuming that there is no slack in the string.

The angle of elevation of the top of a tower from a point on the ground is ${45}^{°}$. Moving $21 \mathrm{m}$ directly towards the base of the tower, the angle of elevation changes to ${60}^{°}$.  What is the height of the tower to the nearest meter?

If $\sin (3\mathrm{\alpha }-\mathrm{\beta })=1$ and $\cos (2\mathrm{\alpha }+\mathrm{\beta })=\frac{1}{2}$, then the value of$\tan \mathrm{\alpha }$is ______.

If $\cos x+{\cos }^{2}x=1$, then what is the value of ${\sin }^{4}x+{\sin }^{2}x$?

If $\left(a^2-b^2\right)\sin \left(\theta \right)+2ab\cos \left(\theta \right)=a^2+b^2$, then what is the value of $\tan \left(\theta \right)$?

At a point $P$ on the ground, the angle of elevation of the top of a $10\mathrm{m}$ tall building and of a helicopter hovering some distance over the top of the building are $30º$ and $60º$ respectively. Then, the height of the helicopter above the ground is