Expanding and Simplifying Trigonometric Expressions

The question asks us to find the equivalent expression for $(1 - \sin A + \cos A)^2$. We need to expand this expression and simplify it using trigonometric identities.

Step-by-Step Expansion of $(1 - \sin A + \cos A)^2$

We can expand the given expression $(1 - \sin A + \cos A)^2$ using the algebraic identity for the square of a trinomial:

$\qquad (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$

In our expression, let $a = 1$, $b = -\sin A$, and $c = \cos A$. Substituting these values into the identity, we get:

$\qquad (1 - \sin A + \cos A)^2 = (1)^2 + (-\sin A)^2 + (\cos A)^2 + 2(1)(-\sin A) + 2(1)(\cos A) + 2(-\sin A)(\cos A)$

Let's simplify each term:

• $(1)^2 = 1$
• $(-\sin A)^2 = \sin^2 A$
• $(\cos A)^2 = \cos^2 A$
• $2(1)(-\sin A) = -2\sin A$
• $2(1)(\cos A) = 2\cos A$
• $2(-\sin A)(\cos A) = -2\sin A \cos A$

Combining these terms, the expansion becomes:

$\qquad (1 - \sin A + \cos A)^2 = 1 + \sin^2 A + \cos^2 A - 2\sin A + 2\cos A - 2\sin A \cos A$

Using the Pythagorean Identity

Recall the fundamental trigonometric identity:

$\qquad \sin^2 A + \cos^2 A = 1$

Substitute this into our expanded expression:

$\qquad (1 - \sin A + \cos A)^2 = 1 + (\sin^2 A + \cos^2 A) - 2\sin A + 2\cos A - 2\sin A \cos A$

$\qquad (1 - \sin A + \cos A)^2 = 1 + 1 - 2\sin A + 2\cos A - 2\sin A \cos A$

$\qquad (1 - \sin A + \cos A)^2 = 2 - 2\sin A + 2\cos A - 2\sin A \cos A$

Factoring the Simplified Expression

Now, we need to factor the expression $2 - 2\sin A + 2\cos A - 2\sin A \cos A$ to match one of the given options. We can factor out the common term 2:

$\qquad 2(1 - \sin A + \cos A - \sin A \cos A)$

Let's rearrange the terms inside the parenthesis and try factoring by grouping:

$\qquad 1 - \sin A + \cos A - \sin A \cos A = (1 - \sin A) + (\cos A - \sin A \cos A)$

Factor out $\cos A$ from the second group:

$\qquad (1 - \sin A) + \cos A (1 - \sin A)$

Now, we see that $(1 - \sin A)$ is a common factor:

$\qquad (1 - \sin A)(1 + \cos A)$

Substituting this back into the expression with the factor of 2:

$\qquad 2(1 - \sin A)(1 + \cos A)$

Comparing with Options

Let's compare our final factored expression with the given options:

1. $2(1 – \cos A)(1 + \sin A)$
2. $2(1 – \sin A)(1 + \cos A)$
3. $2(1 – \cos A)(1 – \sin A)$
4. None of the above

Our simplified and factored expression $2(1 - \sin A)(1 + \cos A)$ exactly matches Option 2.

Alternatively, we could expand the correct option $2(1 - \sin A)(1 + \cos A)$ to see if it matches the expanded form of the original expression:

$\qquad 2(1 - \sin A)(1 + \cos A) = 2 [1(1 + \cos A) - \sin A(1 + \cos A)]$

$\qquad = 2 [1 + \cos A - \sin A - \sin A \cos A]$

$\qquad = 2 + 2\cos A - 2\sin A - 2\sin A \cos A$

Rearranging the terms: $2 - 2\sin A + 2\cos A - 2\sin A \cos A$. This matches the expanded form we obtained from $(1 - \sin A + \cos A)^2$, confirming our result.

Conclusion

The expansion and simplification of $(1 - \sin A + \cos A)^2$ leads to $2(1 - \sin A)(1 + \cos A)$.

Revision Table: Key Trigonometry Concepts

Additional Information: Working with Trigonometric Identities

When solving problems involving trigonometric expressions, remember these tips:

• Always look for opportunities to use fundamental identities like $\sin^2 \theta + \cos^2 \theta = 1$, $\tan \theta = \sin \theta / \cos \theta$, $\sec \theta = 1 / \cos \theta$, etc.
• Algebraic techniques such as expansion, factoring, finding a common denominator, and rationalizing can be very useful.
• If the expression involves squares, consider using squared identities or the Pythagorean identity.
• Sometimes, converting everything to sine and cosine functions can simplify the expression.
• If the expression is complicated, try manipulating the options as well to see if they match the original expression after simplification or expansion.