Proving the Trigonometric Identity: sinx / (1 cosx) (1 - cosx) / sinx 2sinx / (1 cosx)

Understanding and proving trigonometric identities is a fundamental skill in mathematics, essential for both academic and professional purposes. In this article, we will explore the step-by-step process to prove the identity:

Step-by-Step Proof

To start, let's restate the identity we need to prove:

left(frac{sin{x}}{1 cos{x}} frac{1 - cos{x}}{sin{x}} frac{2sin{x}}{1 cos{x}}right)

This can be simplified and proven through a series of algebraic manipulations. Here’s how it can be done:

Multiplying Each Term by a Common Denominator

The process begins by multiplying each term by a common denominator, which in this case is sin{x}(1 cos{x}). This step helps in combining the fractions into a single expression.

left(frac{sin{x}}{1 cos{x}} times frac{sin{x}}{sin{x}} frac{1 - cos{x}}{sin{x}} times frac{1 cos{x}}{1 cos{x}} frac{2sin{x}}{1 cos{x}}right)

Combining the Fractions

After multiplying by the common denominator, the fractions can be combined:

left(frac{sin^2{x} (1 - cos{x})(1 cos{x})}{sin{x}(1 cos{x})} frac{2sin{x}}{1 cos{x}}right)

Next, simplify the numerator:

left(frac{sin^2{x} 1 - cos^2{x}}{sin{x}(1 cos{x})} frac{2sin{x}}{1 cos{x}}right)

Recall the trigonometric identity: sin^2{x} cos^2{x} 1. Therefore, we have:

1 - cos^2{x} sin^2{x}

Substituting this into the equation:

left(frac{sin^2{x} sin^2{x}}{sin{x}(1 cos{x})} frac{2sin{x}}{1 cos{x}}right)

Final Simplification

The numerator simplifies to:

2sin^2{x}

This gives:

left(frac{2sin^2{x}}{sin{x}(1 cos{x})} frac{2sin{x}}{1 cos{x}}right)

Cancel out 2sin{x} from the numerator and the bottom part:

left(frac{2sin{x}}{1 cos{x}} frac{2sin{x}}{1 cos{x}}right)

Conclusion

The identity has been proven, demonstrating the power of algebraic manipulation and trigonometric identities.

Key Points

Multiplying by the common denominator to combine fractions. Using trigonometric identities to simplify the expression. Algebraic simplification to reach the desired result.

Additional Resources

For more practice and insights into trigonometric identities, you can refer to the following resources:

Educational videos on trigonometry on YouTube. Online math courses such as Khan Academy or Coursera. Math textbooks specifically focused on trigonometric identities.