Proving the Trigonometric Identities for tan(α) and tan(α - β)

In this article, we will delve into the process of proving two important trigonometric identities involving the tangent function. We will explore the steps to prove that tan(α) tan(α)tan(β) / (1 - tan(α)tan(β)) and tan(α - β) (tan(α) - tan(β)) / (1 tan(α)tan(β)).

1. Proving tan(α) tan(α)tan(β) / (1 - tan(α)tan(β))

Consider a right angle triangle CDE with a unit base length. We start by defining the sides and angles of this triangle:

Let CE be the adjacent side and DE be the opposite side to angle α.

The cosine of angle α is given by:

cos α 1 / CE

Which implies:

CE 1 / cos α

The tangent of angle α is given by:

tan α DE / 1

Which implies:

DE tan α

Now, consider another right angle triangle CEF:

In triangle CEF, we have:

BD is perpendicular to CD, and EF is perpendicular to CE

Therefore, angle FEB α.

The tangent of angle β in this triangle is given by:

tan β EF / (1 / cos α)

Which implies:

EF tan β / cos α

2. Proving tan(α - β) (tan(α) - tan(β)) / (1 tan(α)tan(β))

Now, let's move on to the second identity. In triangle EBF, we have:

The cosine of angle α in triangle EBF is given by:

cos α BE / (tan β / cos α)

Which implies:

BE tan β

The tangent of angle β in triangle EBF is given by:

tan α BF / tan β

Which implies:

BF tan α tan β

Now, consider triangle AFC:

In triangle AFC, we have:

Angle AFC α - β because AB is parallel to CD.

Therefore, the tangent of angle α - β is given by:

tan(α - β) (tan α tan β) / (1 - tan α tan β)

When β -β, the identity becomes:

tan α - β (tan α - tan β) / (1 tan α tan β)

This implies:

tan α - β (tan α - tan β) / (1 tan α tan β)

Conclusion

In summary, we have successfully proven the trigonometric identities for tan(α) and tan(α - β). These proofs are essential in the field of trigonometry and are widely used in various applications, including calculus, physics, engineering, and beyond. Understanding these identities can greatly enhance your problem-solving skills in mathematics and related fields.