Question

if A+B=45 , then prove
(1+tanA)(1+tanB)=2

Answer 1

take tan on both side

Answer 2

If A+B=45, then what is the value of (1+tanA) (1+tanB)? Recall that the tangent of the sum of two angles can be expressed as:  tan(x+y)=tan(x)+tan(y)1−tan(x)tan(y)  In your case, x=a, and y=b.  Then:  tan(a+b)=tan(a)+tan(b)1−tan(a)tan(b)   Since a+b=45º, we have that tan(a+b)=1.  Therefore: tan(a)+tan(b)1−tan(a)tan(b)=1 and thus tan(a)+tan(b)=1−tan(a)tan(b).  Adding 1+tan(a)tan(b) to both sides:  1+tan(a)+tan(b)+tan(a)tan(b)=2  Factoring: (1+tan(a))(1+tan(b))=2,