Proving the Trigonometric Identity: cosA/sinA sinA/(1-cosA) 1/sinA

In this article, we will follow a step-by-step approach to prove the trigonometric identity: cosA/sinA sinA/(1-cosA) 1/sinA. Our task is to show that the left-hand side (LHS) simplifies to the right-hand side (RHS) using standard trigonometric identities.

Step-by-Step Solution

Let's begin by setting up the identity and addressing the left-hand side (LHS).

1. Setting Up the Identity

Given the identity:

cosA/sinA sinA/(1-cosA) 1/sinA

LHS: Partial Fraction Addition

First, we add the fractions on the LHS:

cosA/sinA sinA/(1-cosA)

This can be rewritten as:

(cosA(1-cosA) sinA(sinA))/(sinA(1-cosA)) (cosA 1-cosA sin^2 A)/(sinA(1-cosA))

2. Simplifying the Numerator

The numerator can be simplified using the Pythagorean identity, sin^2 A cos^2 A 1:

1-cos^2 A sin^2 A

Thus, the expression becomes:

cosA(1-cosA) sin^2 A cosA - cos^2 A sin^2 A cosA - cos^2 A 1 - cos^2 A cosA - 2cos^2 A 1

3. Simplifying the Entire Expression

Now, the LHS becomes:

(cosA 2cos^2 A 1)/(sinA(1-cosA))

Using the identity 1 - cosA in the denominator, we observe:

(cosA 2cos^2 A 1)/(sinA(1-cosA)) (cosA 1)/(sinA(1-cosA))

4. Further Simplification

Notice that (cosA 1)/(1-cosA) can be simplified as:

cosA/(1-cosA) -cosA/(cosA-1)

Therefore:

(cosA/(1-cosA)) * (1/(sinA)) 1/(sinA)

Conclusion: Q.E.D.

Thus, we have shown that the LHS reduces to the RHS, proving the identity:

cosA/sinA sinA/(1-cosA) 1/sinA

The steps above confirm that our initial assumption is correct, and we have demonstrated that:

cosA/sinA sinA/(1-cosA) 1/sinA

Q.E.D.

Alternative Method: Cross-Multiplication

An alternative method to prove the same identity involves cross-multiplication:

1. Cross-Multiplying the Identity

Starting with the given identity:

(cosA/sinA) (sinA/(1-cosA)) 1/sinA

We cross-multiply:

cosA(1-cosA) sinA(sinA) sinA(1-cosA)

2. Simplifying the Equations

This simplifies to:

cosA - cos^2 A sin^2 A sinA - sinAcosA

Using the Pythagorean identity, we know that:

1 - cos^2 A sin^2 A

Hence:

cosA sin^2 A sinA - sinAcosA

Since both sides simplify to the same expression, we have proven the identity.

Q.E.D.

Conclusion

By following these detailed steps, we have successfully proven the given trigonometric identity using both methods of partial fraction addition and cross-multiplication. This identity is a useful tool in simplifying and solving trigonometric expressions. For more information on similar trigonometric identities and their proofs, please refer to the resources provided below.

Keywords:

Trigonometric identities, proving trigonometric identities, proving the given identity