Final answer:
To express 1 - tan(a) as a product, utilize the trigonometric identity for tan(a + b) and equate it with tan(2a), which ultimately allows factoring the expression into a product involving tan(2a).
Step-by-step explanation:
To express 1 - tan(a) as a product, we need to use trigonometric identities. There is a well-known identity involving the tangent function:
tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b))
If we compare this with the desired expression, we can let tan(b) = tan(a), then we will get:
tan(a + a) = (tan(a) + tan(a)) / (1 - tan(a)tan(a))
Simplify to find:
tan(2a) = 2tan(a) / (1 - tan^2(a))
Thus:
1 - tan^2(a) = 2tan(a) / tan(2a)
From the above equation, we can now express our original expression 1 - tan(a) in terms of tan(2a) by factoring:
1 - tan(a) = (1 - tan(a)) * (1 + tan(a)) / (1 + tan(a))
= (1 - tan^2(a)) / (1 + tan(a))
= 2tan(a) / (tan(2a) * (1 + tan(a)))
We have now successfully expressed 1 - tan(a) as a product.