Given that tan^2x=1-a^2tan2x=1−a2
rarrsecx=sqrt(1+tan^2x)=sqrt(1+1-a^2)=sqrt(2-a^2)=(2-a^2)^(1/2)→secx=√1+tan2x=√1+1−a2=√2−a2=(2−a2)12
LHS=secx+tan^3x*cscxLHS=secx+tan3x⋅cscx
=secx+cancel(sinx)/cosx*(sin^2x/cos^2x)*1/cancel(sinx)
=secx+sec^3x*sin^2x=secx(1+sec^2x*sin^2x)
=secx(1+sin^2x/cos^2x)=secx*[(sin^2x+cos^2x)/cos^2x]
=sec^3x=((2-a^2)^((1/2)))^3=(2-a^2)^(3/2)=RHS