Problem 55

Question

Perform the multiplication and use the fundamental identities to simplify. $$(\sin x+\cos x)^{2}$$

Step-by-Step Solution

Verified

Answer

The simplified form of the given expression $(\sin x + \cos x)^2$ is \[ 1 + \sin(2x) \].

1Step 1: Expand the expression

Expand the entire expression by squaring the terms: \[ (\sin x + \cos x)^2 = (\sin x)^2 + 2 \sin x \cos x + (\cos x)^2 \]. This is equivalent to using the formula: $(a+b)^2=a^2+2ab+b^2$.

2Step 2: Apply trigonometric identities

Using the identity \[ \sin^2 x + \cos^2 x = 1 \], and also recognizing that \[2\sin x \cos x\] is the double angle identity for sine (i.e, \[2\sin x \cos x = \sin(2x)\]), the equation can be rewritten as \[1 + \sin(2x)\].

3Step 3: Final simplification

Since we have replaced all terms with their respective identities, no further simplification is required. So, the answer is \[1 + \sin(2x)\].

Key Concepts

Expanding ExpressionsDouble Angle IdentityThe Identity $\sin^2 x + \cos^2 x = 1$

Expanding Expressions

When we talk about expanding expressions in mathematics, it generally means transforming a concise formula into a more detailed form by multiplication, addition, or other algebraic operations. In our original problem, we need to expand $(\sin x + \cos x)^2$. This is a classic example of squaring a binomial, a common type of expression in algebra, and it requires applying the formula

$(a+b)^2 = a^2 + 2ab + b^2$

In this context, $a$ is $\sin x$ and $b$ is $\cos x$. By substituting these values into the formula, we perform the steps:

Square each term: $(\sin x)^2 + (\cos x)^2$
Add twice the product of both terms: $2 \sin x \cos x$

So, expanding $(\sin x + \cos x)^2$ gives us \[(\sin x)^2 + 2 \sin x \cos x + (\cos x)^2\]. This expansion lays the groundwork for applying trigonometric identities for simplification. Breaking down expressions in this way is crucial before moving on to further simplification.

Double Angle Identity

The double angle identity is a key trigonometric formula that expresses trigonometric functions of double angles (like $2x$) in terms of single angle functions (like $x$). In the step-by-step solution, we encounter the term $2 \sin x \cos x$, which is a perfect example for using this identity. The double angle identity for sine is

\[2 \sin x \cos x = \sin(2x)\]

This identity simplifies our expanded expression and allows us to rewrite two terms as one. Recognizing and applying such identities helps make complex trigonometric expressions more manageable. Here, it reduces the number of terms we work with and should always be on your radar as you follow through with simplifications.Using identities like the double angle identity, we can efficiently move from the expanded form \[(\sin x)^2 + 2 \sin x \cos x + (\cos x)^2\] to a more simplified expression \[1 + \sin(2x)\].This highlights the strategic advantage of mastering trigonometric identities.

The Identity $\sin^2 x + \cos^2 x = 1$

The trigonometric identity $\sin^2 x + \cos^2 x = 1$ is an essential cornerstone of trigonometry. It originates from the Pythagorean theorem applied to a unit circle. Here, when $x$ is an angle, the coordinates $(\cos x, \sin x)$ lie on the circle with radius 1.In our problem, after expanding $(\sin x + \cos x)^2$, you observe the terms $(\sin x)^2$ and $(\cos x)^2$. By employing the identity

\[\sin^2 x + \cos^2 x = 1\]

these two terms combine and simplify to 1. This is a vital step that enables major simplification in trigonometric calculations and reduces cumbersome expressions into elegant, easy-to-digest forms.Replacing the sum of these squares with 1 in our expression gives it the simplified form \[1 + \sin(2x)\]. Whenever you see the terms $(\sin x)^2$ and $(\cos x)^2$ together without any other operations between them, remember this identity, as it will save numerous steps in your calculations.

Problem 55