Question

tantheta+sintheta= m , tantheta-sintheta=n , then prove that msquare-nsquare=4underrootmn

Answer 1

Tan theta +sin theta =m......1
Tan theta -sin theta =n........2
Solving, LHS
On squaring and subtracting both the equipment
=(tan theta +sin theta)^2- (tan theta -sin theta)^2= m^2 -n^2
=tan ^2A + sin ^2A +2tan A sin A -Tan^2 A -sin^2 A +2tan A sin A= m^2-n^2
=4tan A sin A=m^2 -n^2
Now, RHS
4root mn=4root (tan A+sin A)(tan A-sin A)
4root tan^2 A-sin ^2A. (since (a+b) (a-b) = a ^2-b^2)
4root sin^2 A/cos^2 A- sin^2 A
4root sin^2 A(1-cos^2 A)/ cos ^2A.
4root tan^2 A sin ^2A
4tan A sin A.
m^2-n^2
So, LHS=RHS