Question

Prove that :
$(sinA+secA{)}^2+(cosA+cosecA{)}^2=(1+secAcosecA{)}^2$

Answer 1

We have,

$LHS=(sinA+secA{)}^2+(cosA+cosecA{)}^2$

$\Rightarrow LHS=si{n}^{2}A+se{c}^{2}A+2sinAsecA+co{s}^{2}A+cose{c}^{2}A+2cosAcosecA$

$\Rightarrow LHS=(si{n}^{2}A+co{s}^{2}A)+(se{c}^{2}A+cose{c}^{2}A)+2sinAsecA+2cosAcosecA$

$\Rightarrow LHS=1+(\frac{1}{co{s}^{2}A}+\frac{1}{si{n}^{2}A})+2(\frac{sinA}{cosA}+\frac{cosA}{sinA})$

$\Rightarrow LHS=1+\left(\frac{si{n}^{2}A+co{s}^{2}A}{si{n}^{2}Aco{s}^{2}A}\right)+2\left(\frac{si{n}^{2}A+co{s}^{2}A}{sinAcosA}\right)$

$\Rightarrow LHS=1+\frac{1}{si{n}^{2}Aco{s}^{2}A}+\frac{2}{sinAcosA}$

$\Rightarrow LHS=1+cose{c}^{2}Ase{c}^{2}A+2cosecAsecA=(1+secAcosecA{)}^2=RHS$