2011/019) Prove that Sin (a +b ) sin (a-b) = sin ^2 a – sin ^2b

The above is an interesting identify in trigonometry and this can be done wrongly with out understanding trigonometry

As we do not know trigonometry we take sin as a multiplier

So sin (a+b) = sin a + sin b

And sin (a-b) = sin a – sin b

Multiply them to get

Sin (a+b) sin (a-b) = (sin a + sin b) (sin a - sin b)
= sin ^2 a – sin ^2 b

however above is wrong but the ans is right

now to the correct approach

Sin (a+b) sin (a-b)
= (sin a cos b + cos a sin b)(sin a cos b – cos a sin b)
= sin ^2 a cos^2b – cos^2 a sin ^2b
= sin ^2 a(1- sin ^2 b) – cos^2 a sin ^2 b
= sin ^2 a – sin ^2 a sin ^2 b – cos^2 a sin^2 b
= sin ^2 a – sin ^2 b(sin ^2 a + cos^2 a)
= sin ^2 a – sin ^2 b


The above example is to demonstrate that we get right result through erroneous approach and one need to be careful about it