Proving the Trigonometric Identity: (frac{1-costhetasintheta}{1-costheta-sintheta} frac{1}{sintheta / costheta})

In this article, we will explore how to prove the identity (frac{1-costhetasintheta}{1-costheta-sintheta} frac{1}{sintheta / costheta}). This proof involves several steps, including the use of trigonometric identities and algebraic manipulation. By following the detailed steps outlined below, you will gain a deeper understanding of how to approach and solve similar trigonometric problems.

Step-by-Step Proof

Let's start by using the fundamental identity (cos^2theta sin^2theta 1). We aim to manipulate the left side of the equation to show that it simplifies to the right side. Here is a detailed step-by-step proof:

Start with the left side of the equation:

[frac{1-costhetasintheta}{1-costheta-sintheta}]

From the fundamental identity, we know that:

[cos^2theta 1 - sin^2theta]

Therefore, we can rewrite the numerator:

[1 - costhetasintheta 1 - costhetasintheta costhetacostheta - costhetacostheta]

[ cos^2theta - costhetasintheta 1 - sin^2theta]

[ costheta(costheta - sintheta) (1 - sin^2theta)]

Now, let's simplify the denominator:

[1 - costheta - sintheta]

Notice that:

[1 - costheta - sintheta 1 - sintheta - costheta]

Rationalize the expression by factoring out common terms:

[frac{costheta(costheta - sintheta) (1 - sin^2theta)}{1 - costheta - sintheta}]

Use the algebraic identity ((1 - sintheta)(1 sintheta) 1 - sin^2theta):

[frac{costheta(costheta - sintheta) (1 - sin^2theta)}{(1 - sintheta)(1 sintheta)}]

Now, observe that the numerator can be written as:

[costheta(costheta - sintheta) (1 - sin^2theta) costheta(costheta - sintheta) (1 - sintheta)(1 sintheta)]

Factor the expression:

[costheta(costheta - sintheta) (1 - sintheta)(1 sintheta) (1 - sintheta)(costheta 1 sintheta)]

Combine the factored terms in the numerator and the denominator:

[frac{(1 - sintheta)(costheta 1 sintheta)}{(1 - sintheta)(1 sintheta)}]

[frac{(1 - sintheta)(costheta 1 sintheta)}{(1 - sintheta)(1 sintheta)} frac{costheta 1 sintheta}{1 sintheta}]

Divide by (sintheta / costheta):

[frac{1}{frac{sintheta}{costheta}} frac{costheta}{sintheta}]

Therefore, we have shown that:

[frac{1-costhetasintheta}{1-costheta-sintheta} frac{costheta}{sintheta} frac{1}{sintheta / costheta}]

Conclusion

In conclusion, we have successfully proven the identity using basic trigonometric identities and algebraic manipulations. By following these steps, you can solve similar trigonometric problems by breaking down complex expressions step-by-step. Understanding the fundamental identities and practicing with various examples will greatly enhance your problem-solving skills in trigonometry.

Additional Reading and Resources

To further deepen your understanding of trigonometric identities and proof techniques, consider exploring the following resources:

Math is Fun: Trigonometric Identities Khan Academy: Trigonometric Identities Review Lamar University: Trigonometric applications

By utilizing these resources, you can gain a more comprehensive understanding of trigonometry and prepare for advanced mathematical concepts.