Proving Trigonometric Identities: cosA/(1-sinA) (1-sinA)/cosA

To prove the equation frac{cos A}{1 - sin A} frac{1 - sin A}{cos A}, we can start by cross-multiplying to eliminate the fractions. This yields:

Step 1: Cross-Multiplication

Cross-multiplying the given equation gives:

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

Step 2: Simplify the Right-Hand Side

The right-hand side can be simplified using the difference of squares:

cos^2 A 1 - 2sin A sin^2 A

Using the Pythagorean identity:

cos^2 A sin^2 A 1

We can express the equation as:

1 - sin^2 A cos^2 A

Step 3: Equate the Simplified Form

Substituting this back into our original equation gives us:

cos^2 A cos^2 A

As both sides are now equal, this proves that:

frac{cos A}{1 - sin A} frac{1 - sin A}{cos A}

Alternative Proofs

Proof 1

Let's consider another proof by LHS manipulation:

LHS frac{cos A}{1 - sin A} frac{cos A(1 sin A)}{(1 - sin A)(1 sin A)} frac{cos A (1 sin A)}{1 - sin^2 A} frac{cos A (1 sin A)}{cos^2 A} frac{1 sin A}{cos A} RHS

Proof 2

Another verification involves a step-by-step breakdown:

1 - sin A cos A / (1 - sin A)

Multiplying both sides by cos A in the numerator and denominator gives:

frac{1 - sin A}{cos A} frac{cos A}{1 - sin A}

Proof 3

Let's use a different method involving manipulating the given expression:

Define:

x frac{1 - sin A}{cos A} frac{1 - sin A}{1 - sin A}

By Dividendo:

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

By Componendo:

frac{x 1}{1} frac{1 sin A}{cos A}

Substituting back, we get:

frac{1 - sin A}{cos A} frac{1 - sin A}{cos A}

Validity of the Identity

Note that the identity holds true as long as cos A eq 0 and 1 - sin A eq 0. If cos A 0 text{ or } sin A -1, the identity does not hold due to undefined or zero denominators. Specifically, this occurs when A kfrac{pi}{2}, where k is an odd integer.