Difficulty proving another identity

[rhule009]

could anyone explain (no calculation) how to work the following identity so could get an understanding of it thank you

tan(45+A)degrees tan(45-A)degrees=1

 

[neutrino]

Use the identities for tan(a+b) and tan(a-b).

 

[Architeuthis Dux]

To understand this identity, it is helpful to first recall the trigonometric identities for the sum and difference of two angles:

1. sin(A+B) = sin(A)cos(B) + cos(A)sin(B)
2. cos(A+B) = cos(A)cos(B) - sin(A)sin(B)
3. tan(A+B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))

Now, let's apply these identities to the given equation:

tan(45+A)tan(45-A) = (tan(45) + tan(A)) / (1 - tan(45)tan(A)) * (tan(45) - tan(A)) / (1 + tan(45)tan(A))

We can simplify this further by using the fact that tan(45) = 1:

(tan(A) + 1) / (1 - tan(A)) * (1 - tan(A)) / (1 + tan(A))

Now, notice that the (1 - tan(A)) terms cancel out, leaving us with:

tan(A) + 1 = tan(A) + 1

This shows that the original equation is indeed true, as both sides are equal to tan(A) + 1. This may seem like a simple proof, but it relies on the fundamental trigonometric identities and understanding of how to manipulate and simplify equations using those identities.

In conclusion, to fully understand this identity, it is important to have a strong grasp of trigonometric concepts and identities. Without this foundation, it may be difficult to prove the identity or fully understand it.