Proving the Trigonometric Identity: 1 - sin A - cos A2 2(1 - sin A)(1 - cos A)

In trigonometry, proving identities is a fundamental skill that helps us understand the relationships between different trigonometric functions. Today, we will delve into the proof of the identity 1 - sin A - cos A2 2(1 - sin A)(1 - cos A). This step-by-step guide aims to clearly demonstrate how to derive and prove this identity using basic trigonometric properties and identities.

Proof Steps

Step 1: Expanding the Left Side

To start, we expand the left side of the identity (1 - sin A - cos A2):

1 - sin A - cos A2 Using the distributive property (FOIL), we get: 1 - sin A - cos A (1 - sin A - cos A) This can be expanded as 1 - 2sin A - 2cos A sin2 A 2sin A cos A cos2 A

Step 2: Simplification Using Identities

Next, we simplify using the Pythagorean identity sin2 A cos2 A 1:

1 - sin A - cos A (1 - sin A - cos A) Simplifying the expression, we get: 1 - 2sin A - 2cos A 1 2sin A cos A This simplifies to 2 - 2sin A - 2cos A 2sin A cos A

Step 3: Factoring the Left Side

We factor out a common factor of 2 from the left side:

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

Step 4: Expanding the Right Side

Next, we expand the right side of the identity (2(1 - sin A)(1 - cos A)):

2(1 - sin A)(1 - cos A) Using the distributive property, we get: 2(1 - cos A - sin A sin A cos A)

Step 5: Comparing Both Sides

Now we compare both sides:

Left side: 2(1 - sin A - cos A sin A cos A) Right side: 2(1 - cos A - sin A sin A cos A)

We see that both sides are equal. Therefore, we have proven the identity:

1 - sin A - cos A2 2(1 - sin A)(1 - cos A)

Conclusion

The identity is proven to be true. This proof demonstrates the power of trigonometric identities in simplifying and proving complex relationships. Understanding and being able to manipulate these identities is crucial in solving more advanced trigonometric problems.