Proving the Trigonometric Identity: tan θ - sin θ / sin3θ
Often, students and enthusiasts encounter questions that seem simple but require a deeper understanding to solve. The expression tan θ - sin θ / sin3θ is one such example. This article aims to guide you through the process of proving this trigonometric identity.
The Expression in Question
The given expression is tan θ - sin θ / sin3θ. You may ask, 'Does this equal something?' Let's explore this and simplify it to see if a simpler expression results.
Understanding Trigonometric Identities
Trigonometric identities are equations that hold true for all values of the variables involved. They are fundamental in simplifying and solving complex trigonometric expressions. The 'dumb question' here is not dumb; it's an invitation to explore and prove the identity.
Breaking Down the Expression
First, let's rewrite the expression in terms of sine and cosine functions. Recall that tan θ sin θ / cos θ.
tan θ - sin θ / sin3θ sin θ / cos θ - sin θ / sin3θ
Combining the Fractions
To combine the fractions, we need a common denominator. The common denominator for the terms is sin3θ cos θ.
sin θ / cos θ - sin θ / sin3θ (sin θ sin2θ - sin θ) / (sin3θ cos θ)
Now, let's simplify the numerator:
(sin θ sin2θ - sin θ) / (sin3θ cos θ) (sin θ (sin2θ - 1)) / (sin3θ cos θ)
Simplifying Further
Recall the Pythagorean identity sin2θ cos2θ 1. Therefore, sin2θ - 1 -cos2θ.
(sin θ (sin2θ - 1)) / (sin3θ cos θ) (sin θ (-cos2θ)) / (sin3θ cos θ)
Final Simplification
Now, let's simplify the fraction:
(sin θ (-cos2θ)) / (sin3θ cos θ) -cos2θ / sin2θ
This simplifies to:
-cot2θ
Conclusion
We have successfully proven the identity:
tan θ - sin θ / sin3θ -cot2θ
Further Exploration
Understanding and proving trigonometric identities is essential for solving complex problems in mathematics and physics. It improves problem-solving skills and enhances your mathematical intuition.