Question

State true or false:

$\cos \theta =\frac{a^2{b}^2}{2ab},$ where a and b are two distinct numbers such that $ab>0.$

Answer 1

The value of $\cos \theta$ varies bwtween $0$ & $1.$

If $\cos \theta =0$, then $\frac{a^2{b}^2}{2ab}=0⟹ab=0$

But, $ab\ne 0.$

So, $\frac{a^2{b}^2}{2ab}\ne 0.$

$\therefore \frac{a^2{b}^2}{2ab}=1$ (at best).

$⟹ab=2=2\times 1.$

Since no other factor of 2 is possible,

$(a,b)=(1,2)$ or $(2,1)$ which are distinct numbers.

$\therefore$ The assertion is true.