Question

If x = tanA + sinA and y = tanA - sinA , show thatx 2 - y 2 = 4 (root xy)

Answer 1

tanA+sinA=x and tanA-sinA=y
∴, x²-y²
=(x+y)(x-y)
=(tanA+sinA+tanA-sinA)(tanA+sinA-tanA+sinA)
=(2tanA)(2sinA)
=4tanAsinA
4√xy
=4√(tanA+sinA)(tanA-sinA)
=4√(tan²A-sin²A)
=4√{(sin²A/cos²A)-sin²A}
=4√sin²A{(1/cos²A)-1}
=4sinA√{(1-cos²A)/cos²A}
=4sinA√(sin²A/cos²A)
=4sinA√tan²A
=4sinAtanA
∴, LHS=RHS (Proved)