Question

sin a/1 plus cos a plus 1 plus cos a/sina multiplied by cos a/1 minus sin a minus 1 minus sin a/cos a is equal to 4sec a.

Answer 1

$$\begin{gathered} \left(\frac{\mathrm{sina}}{1+\mathrm{cosa}}+\frac{1+\mathrm{cosa}}{\mathrm{sina}}\right)\left(\frac{\mathrm{cosa}}{1-\mathrm{sina}}-\frac{1-\mathrm{sina}}{\mathrm{cosa}}\right)=4\mathrm{seca} \\ \mathrm{LHS}, \\ \frac{{\sin }^2\mathrm{a}+1+{\cos }^2\mathrm{a}+2\mathrm{cosa}}{\mathrm{sina}\left(1+\mathrm{cosa}\right)}\times \frac{{\cos }^2\mathrm{a}-1-{\sin }^2\mathrm{a}+2\mathrm{sina}}{\mathrm{cosa}\left(1-\mathrm{sina}\right)} \\ =\frac{2\left(1+\mathrm{cosa}\right)}{\mathrm{sina}\left(1+\mathrm{cosa}\right)}\times \frac{1-{\sin }^2\mathrm{a}-1-{\sin }^2\mathrm{a}+2\mathrm{sina}}{\mathrm{cosa}\left(1-\mathrm{sina}\right)} \\ =\frac{2}{\mathrm{sina}}\times \frac{2\mathrm{sina}\left(1-\mathrm{sina}\right)}{\mathrm{cosa}\left(1-\mathrm{sina}\right)}=4\mathrm{seca} \end{gathered}$$