Question

If tanA + sinA = m and tanA - sinA = n. Then show that m sq. - n sq. = +- 4√mn

Answer 1

L. H. S
Squaring both sides,
(tanA+sin A)2=m2

Or, Tan2A+sin2A+2tanAsinA =m2..(i).

Similarly,
(TanA-sinA) 2=n2

Or,tan2A+sin2A-2tanAsinA=n2....(ii)
(*using formula (a+b)2 and (a-b) 2 in the above eqs.)

(i) - (ii)
4tanAsinA=m2-n2

RHS
+-4 root mn=+-root (tanA+sin A) (TanA-sinA)

=root(tan2A-sin2A)........ ( a2-b2 formula)

=root (sin2A/cos2A-sin2A)....

**Note: (tan2A=sin2A/cos2A)

=root[(sin2A(1/cos2A-1).. (taking sin2A common)

=Root [sin2A(sec2A-1)].......... (1/cosA=secA)

=Root(sin2A tan2A)......] [(sec2A-1)=tan2A]
=sinA. tanA.
Therefore, LHS=RHS(PROVED)