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Question Number 54239 by 951172235v last updated on 01/Feb/19

Answered by Prithwish sen last updated on 01/Feb/19

= tan^(−1) ((((1/(p+q+s)) +(1/(p+q+t)))/(1− (1/((p+q+s)(p+q+t)))))) +  tan^(−1) ((((1/(p+r+u)) +(1/(p+r+v)))/(1−(1/((p+r+u)(p+r+v))))) )  = tan^(−1) (((2p+2q+s+t)/((p+q)^2  + (p+q)(s+t)+st −1)) ) +  tan^(−1) (((2p+2r+u+v)/((p+r)^2 +(p+r)(u+v)+uv −1)))  = tan^(−1) (((2p+2q+s+t)/(2(p+q)^2 +(p+q)(s+t)))) +  tan^(−1) (((2p+2r+u+v)/(2(p+r)^2 +(p+r)(u+v))))  = tan^(−1) ((1/((p+q)))) + tan^(−1) ((1/((p+r))))  = tan^(−1) (((2p+q+r)/(p^2 +pr+pq+qr−1)))  = tan^(−1) (((2p+q+r)/(2p^2 +pq+pr)))  =tan^(−1) ((1/p))  Hence proved.

$$=\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{\frac{\mathrm{1}}{\mathrm{p}+\mathrm{q}+\mathrm{s}}\:+\frac{\mathrm{1}}{\mathrm{p}+\mathrm{q}+\mathrm{t}}}{\mathrm{1}−\:\frac{\mathrm{1}}{\left(\mathrm{p}+\mathrm{q}+\mathrm{s}\right)\left(\mathrm{p}+\mathrm{q}+\mathrm{t}\right)}}\right)\:+ \\ $$$$\mathrm{tan}^{−\mathrm{1}} \left(\frac{\frac{\mathrm{1}}{\mathrm{p}+\mathrm{r}+\mathrm{u}}\:+\frac{\mathrm{1}}{\mathrm{p}+\mathrm{r}+\mathrm{v}}}{\mathrm{1}−\frac{\mathrm{1}}{\left(\mathrm{p}+\mathrm{r}+\mathrm{u}\right)\left(\mathrm{p}+\mathrm{r}+\mathrm{v}\right)}}\:\right) \\ $$$$=\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2p}+\mathrm{2q}+\mathrm{s}+\mathrm{t}}{\left(\mathrm{p}+\mathrm{q}\right)^{\mathrm{2}} \:+\:\left(\mathrm{p}+\mathrm{q}\right)\left(\mathrm{s}+\mathrm{t}\right)+\mathrm{st}\:−\mathrm{1}}\:\right)\:+ \\ $$$$\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2p}+\mathrm{2r}+\mathrm{u}+\mathrm{v}}{\left(\mathrm{p}+\mathrm{r}\right)^{\mathrm{2}} +\left(\mathrm{p}+\mathrm{r}\right)\left(\mathrm{u}+\mathrm{v}\right)+\mathrm{uv}\:−\mathrm{1}}\right) \\ $$$$=\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2p}+\mathrm{2q}+\mathrm{s}+\mathrm{t}}{\mathrm{2}\left(\mathrm{p}+\mathrm{q}\right)^{\mathrm{2}} +\left(\mathrm{p}+\mathrm{q}\right)\left(\mathrm{s}+\mathrm{t}\right)}\right)\:+ \\ $$$$\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2p}+\mathrm{2r}+\mathrm{u}+\mathrm{v}}{\mathrm{2}\left(\mathrm{p}+\mathrm{r}\right)^{\mathrm{2}} +\left(\mathrm{p}+\mathrm{r}\right)\left(\mathrm{u}+\mathrm{v}\right)}\right) \\ $$$$=\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\left(\mathrm{p}+\mathrm{q}\right)}\right)\:+\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\left(\mathrm{p}+\mathrm{r}\right)}\right) \\ $$$$=\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2p}+\mathrm{q}+\mathrm{r}}{\mathrm{p}^{\mathrm{2}} +\mathrm{pr}+\mathrm{pq}+\mathrm{qr}−\mathrm{1}}\right) \\ $$$$=\:\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{2p}+\mathrm{q}+\mathrm{r}}{\mathrm{2p}^{\mathrm{2}} +\mathrm{pq}+\mathrm{pr}}\right) \\ $$$$=\mathrm{tan}^{−\mathrm{1}} \left(\frac{\mathrm{1}}{\mathrm{p}}\right) \\ $$$$\mathrm{Hence}\:\mathrm{proved}. \\ $$

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