#(1)#
#"consider the left hand side"#
#(sinx+cosx)^2+(sinx-cosx)^2#
#"expand factors using FOIL and simplify"#
#=sin^2xcancel(2sinxcosx)+cos^2x+sin^2xcancel(-2sinxcosx)+cos^2x#
#[sin^2x+cos^2x=1]#
#=2(sin^2x+2cos^2x)#
#=2=" right hand side"#
#(2)#
#"consider the left hand side"#
#(2-cos^2x)/(sinx)#
#=(2-(1-sin^2x))/(sinx)#
#=(1+sin^2x)/(sinx)#
#=1/sinx+(sin^2x)/sinx#
#=cscx+sinx=" right hand side"#
#(3)#
#"consider the right hand side"#
#"numerator "=cscx-tanx#
#color(white)("numerator ")=1/sinx-sinx/cos#
#color(white)("numerator ")=(cosx-sin^2x)/(sinxcosx)#
#"denominator "=secx+cotx#
#color(white)("denominator ")=1/cosx+cosx/sinx#
#color(white)("denominator ")=(sinx+cos^2x)/(sinxcosx)#
#"numerator"/"denominator"#
#=(cosx-sin^2x)/(cancel(sinxcosx))xx(cancel(sinxcosx))/(sinx+cos^2x)#
#=(cosx-sin^2x)/(sinx+cos^2x)=" left hand side"#
