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Question Number 122118 by AbdullahMohammadNurusSafa last updated on 14/Nov/20

$$\frac{tanA}{1-cotA}+\frac{cotA}{1-tanA}=secAcosecA+1$$$$\mathit{P}\mathit{l}\mathit{e}\mathit{a}\mathit{s}\mathit{e}\mathit{p}\mathit{r}\mathit{o}\mathit{v}\mathit{e}\mathit{i}\mathit{t}\mathit{f}\mathit{o}\mathit{r}\mathit{m}\mathit{e}!!!$$

Answered by MJS_new last updated on 14/Nov/20

$$\frac{t}{1-\frac{1}{t}}+\frac{\frac{1}{t}}{1-t}=\frac{t^2}{t-1}-\frac{1}{t(t-1)}=t+\frac{1}{t}+1=$$$$=\frac{s}{c}+\frac{c}{s}+1=\frac{s^2+c^2}{sc}+1=\frac{1}{sc}+1=$$$$=\csc A\sec A+1$$

Answered by bemath last updated on 14/Nov/20

$$LHS$$$$\frac{\frac{\sin A}{\cos A}}{\frac{\sin A-\cos A}{\sin A}}+\frac{\frac{\cos A}{\sin A}}{\frac{\cos A-\sin A}{\cos A}}$$$$=\frac{\sin A}{\cos A}.\frac{\sin A}{(\sin A-\cos A)}+\frac{\cos A}{\sin A}.\frac{\cos A}{(\cos A-\sin A)}$$$$=\frac{\sin {}^{2}A}{\cos A(\sin A-\cos A)}-\frac{\cos {}^{2}A}{\sin A(\sin A-\cos A)}$$$$=\frac{\sin {}^{3}A-\cos {}^{3}A}{\cos A.\sin A(\sin A-\cos A)}$$$$=\frac{\sin {}^{2}A+\sin A.\cos A+{\cos }^{2}A}{\cos A.\sin A}$$$$=\frac{1+\sin A.\cos A}{\cos A.\sin A}=\sec A.\mathrm{cosec}A+1$$$$$$

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