(c) 1
In a right angled triangle with a, b as sides and c as hypotenuse,
c2 = a2 + b2 (Pythagoras’ Theorem)
Now, given expression = \(\frac{log_{c+b}\,a+log_{c-b}\,a}{2\times\,log_{c+b}\,a\times\,log_{c-b}\,a}\)
= \(\frac{1}{2}\bigg[\frac{1}{\text{log}_{c-b}\,a}+\frac{1}{\text{log}_{c+b}\,a}\bigg]\)
= \(\frac{1}{2}\big[\text{log}_a(c-b)+\text{log}_a(c+b)\big]\) = \(\frac{1}{2}\big[\text{log}_a(c-b)(c+b)\big]\)
= \(\frac{1}{2}\big[\text{log}_a(c^2-b^2)\big]\) = \(\frac{1}{2}\text{log}_a\,a^2\) = loga a = 1.