Answer :

The result tan–1 x – tan–1 is true when value of xy is > -1.


We have,



Principal range of tan-1a is


Let tan-1x = A and tan-1y = B … (1)


So, A,B ϵ


We know that, … (2)


From (1) and (2), we get,



Applying, tan-1 both sides, we get,



As, principal range of tan-1a is .


So, for tan-1tan(A-B) to be equal to A-B,


A-B must lie in – (3)


Now, if both A,B < 0, then A, B ϵ


A ϵ and -B ϵ


So, A – B ϵ


So, from (3),


tan-1tan(A-B) = A-B



Now, if both A,B > 0, then A, B ϵ


A ϵ and -B ϵ


So, A – B ϵ


So, from (3),


tan-1tan(A-B) = A-B



Now, if A > 0 and B < 0,


Then, A ϵ and B ϵ


A ϵ and -B ϵ


So, A – B ϵ (0,π)


But, required condition is A – B ϵ


As, here A – B ϵ (0,π), so we must have A – B ϵ




Applying tan on both sides,



As,


So, tan A < - cot B


Again,


So,


⇒ tan A tan B < -1


As, tan B < 0


xy > -1


Now, if A < 0 and B > 0,


Then, A ϵ and B ϵ


A ϵ and -B ϵ


So, A – B ϵ (-π,0)


But, required condition is A – B ϵ


As, here A – B ϵ (0,π), so we must have A – B ϵ




Applying tan on both sides,



As,


So, tan B > - cot A


Again,


So,


⇒ tan A tan B > -1


⇒xy > -1


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