$ABCD$ is a rectangular field. In one corner $A$ of the field there is a vertical rod. If the angles of elevation of the top of the rod from $B, C$ and $D$ are $\alpha , \beta$ and $\gamma$ then show that, ${\cot }^2\alpha +{\cot }^2\gamma ={\cot }^2\beta$.

If $\frac{{\cos }^{4}x}{{\cos }^{2}y}+\frac{{\sin }^{4}x}{{\sin }^{2}y}=1$ then show that $\frac{{\cos }^{4}y}{{\cos }^{2}x}+\frac{{\sin }^{4}y}{{\sin }^{2}x}=1$

If $x=\mathrm{cosec}\theta -\sin \theta$ and $y=\sec \theta -\cos \theta$ then show that $x^2{y}^2\left(x^2+y^2+3\right)=1$.

If ${\sin }^2\alpha ={\cos }^3\alpha$ then show that ${\cot }^6\alpha -{\cot }^2\alpha =1$.

If $\tan \alpha =p$, then find the value of $\sec \alpha +{\tan }^3\alpha .\mathrm{cosec}\alpha$.

If $\tan \theta =\frac{2n}{n^2-1}$, then show that $\left(n+2\right)\sin \theta +(2n-1)\cos \theta =2n+1$.

A spherical balloon of radius $r$ while bloating in the sky, makes an angle $\alpha$ in the eye of a viewer. If the angle of elevation of the centre of the balloon in the eye of the viewer be $\beta$, show that the altitude of the centre of the balloon from the ground is $r.\mathrm{cosec}\frac{\alpha }{2}.\sin \beta$.

If $7\cos \theta +5\sin \theta =5$, then find the value of $5\cos \theta -7\sin \theta$.

If $\mathrm{cosec\alpha }+\mathrm{cosec\beta }+\mathrm{cosec\gamma }=0$, then show that ${\left(\sin \alpha +\sin \beta +\sin \gamma \right)}^2={\sin }^2\alpha +{\sin }^2\beta +{\sin }^2\gamma$.