Proving Identities in Trigonometry: A Step-by-Step Guide

Understanding and mastering the art of proving trigonometric identities is a fundamental skill in mathematics, particularly in calculus and advanced algebra. This guide will walk you through the process of proving a specific identity, emphasizing the step-by-step method that ensures clarity and accuracy. The identity we will explore is:

sin θ · (1 / tan θ) cos θ · (1 / cot θ) sec θ · csc θ

Step 1: Expand the Left-Hand Side

Let's start with the left-hand side of the equation:

sin θ · (1 / tan θ) cos θ · (1 / cot θ)

Expand this expression:

sin θ · (1 / (sin θ / cos θ)) cos θ · (1 / (cos θ / sin θ))

Step 2: Substitute for tan θ and cot θ

Recall the definitions of tangent and cotangent:

tan θ sin θ / cos θ and cot θ cos θ / sin θ

Substitute these into our expanded expression:

sin θ · (cos θ / sin θ) cos θ · (sin θ / cos θ)

Step 3: Simplify Each Term

Now, let's simplify each term in the expression:

cos θ sin θ

Step 4: Combine the Last Two Terms

We need a common denominator for the terms to combine them:

cos θ · sin θ sin θ · cos θ / sin θ · cos θ (sin^2 θ cos^2 θ) / sin θ · cos θ

Using the Pythagorean identity sin^2 θ cos^2 θ 1, we can simplify the numerator:

1 / sin θ · cos θ

Step 5: Express the Right-Hand Side

Let's rewrite the right-hand side of the equation:

sec θ · csc θ (1 / cos θ) · (1 / sin θ) 1 / (sin θ · cos θ)

Step 6: Final Comparison

Compare the simplified left-hand side and the right-hand side:

left( sin θ cos θ right) / (sin θ · cos θ) 1 / (sin θ · cos θ)

Both sides are equal, confirming the identity:

sin θ · (1 / tan θ) cos θ · (1 / cot θ) sec θ · csc θ

Conclusion

The given identity is thus proven: