Question

Express 1-tan(a) as a product

Answer 1

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.