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If in a right angled triangle, a\ a n d\...

If in a right angled triangle, `a\ a n d\ b` are the lengths of sides and `c` is the length of hypotenuse and `c-b!=1,\ c+b!=1` , then show that `(log)_("c"+"b")"a"+(log)_("c"-"b")=2(log)_("c"+"b")adot(log)_("c"-"b")adot`

Text Solution

Verified by Experts

We know that in a right angled triangle
`c^(2)=a^(2)+b^(2)`
`c^(2)-b^(2)=a^(2)`……….(i)
LHS = `1/(log_(a)(c+b))+1/(log_(a)(c-b)) = (log_(a)(c-b)+log_(a)(c+b))/(log_(a)(c+b).log_(a)(c-b))`
`=log_(a)(c^(2)-b^(2))/(log_(a)(c+b).log_(a)(c-b)) = (log_(a)a^(2))/(log_(a)(c+b).log_(a)(c-b))` ................(using (i))
`=2/(log_(a)(c+b).log_(a)(c-b))=2log_(c+b)a.log_(c-b)a`=RHS
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