Geometric Verification of Identities

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 $\tan \left(\theta \right)$?

If $2x-\frac{1}{2x}=6$, then what is the value of $x^2+\frac{1}{16x^2}$?

If $\frac{{\left(x-y\right)}^2}{x^2-y^2}=9$ then, the value of $\frac{{\displaystyle \frac{x}{y}}+1}{{\displaystyle \frac{x}{y}}-1}$ is __________.

If $x=\frac{2\sqrt{10}}{7}$ then, find the value of $\frac{\sqrt{1+x}-\sqrt{1-x}}{\sqrt{1+x}+\sqrt{1-x}}$.

If $tan\theta +\frac{1}{tan\theta }=2$, then what is the value of $ta{n}^2\left(\theta \right)+\frac{1}{ta{n}^2\left(\theta \right)}$?

If$x+\frac{1}{x}=3$then, what is the value of $x^3+{\left(\frac{1}{x}\right)}^3$?

If $x+\frac{1}{2x}=2$, then what is the value of $\left(8x^3+\frac{1}{x^3}\right)$?

Find the value of the expression $\frac{(469+174{)}^2-(469-174{)}^2}{(469\times 174)}$.

If $x = 2015$,$y = 2014$ and $z = 2013$, then what should come in the place of ($?$) for the below expression?

$x^2+y^2+z^2-xy-yz-zx=?$

If $x=1+\sqrt{2}+\sqrt{3}$ then find the value of $x+\frac{1}{x-1}$.

If $x=2+\sqrt{3}$, then $\left(x^2+\frac{1}{x^2}\right)$ is _____

The angles of elevation of the top of a tower from the top and the foot of a pole of height $10\mathrm{m}$ are $30°$ and $60°$ respectively. Then the height of the tower is...-

If $x = k^3 – 3k^2$ and $y = 1 – 3k$, then find value of $k$, for $x = y$.

If $a^{3}b = abc = 180$. $a, b, c$are positive integers, then the value of $c$ is

What should come in the place of ($?$) for the below expression?

${\left[\frac{\sqrt{2}+1}{2\left(\sqrt{2}-1\right)}-\frac{\sqrt{2}-1}{2\left(\sqrt{2}+1\right)}\right]}^{\frac{1}{3}}=?$

If $x+y+z = 0$, then what should come in the place of ($?$) for the expression given below?

$\frac{{(x+y)}^3+{(y+z)}^3+{(z+x)}^3-17xyz}{10(x+y)(y+z)(z+x)}=?$

If $\mathrm{a}+\mathrm{b}+\mathrm{c}=6$, ${\mathrm{a}}^2+{\mathrm{b}}^2+{\mathrm{c}}^2=14$ and ${\mathrm{a}}^3+{\mathrm{b}}^3+{\mathrm{c}}^3=36$, then the value of $\mathrm{abc}$ is:

If $a+\frac{1}{a} =1$, then find the value of $a^3$.

If $x+y+z=1$, and $xy+yz+zx=-1$, then find the value of $x^2+y^2+z^2$

If $a+b+c=15, abc = 120$ and $(ab+bc+ca)=74$, then the value of $a^3 +b^3 +c^3 =$