Problem 75 - Precalculus Mathematics for Calculus

Question

Prove the identity. $$(\sin x+\cos x)^{2}=1+\sin 2 x$$

Step-by-Step Solution

Answer

The identity \((\sin x + \cos x)^{2} = 1 + \sin 2x\) is proven.

Step 1: Expand the Left Side

Let's start by expanding the left side of the identity: \((\sin x+\cos x)^{2}\). We use the formula for the square of a binomial: \((a+b)^2 = a^2 + 2ab + b^2\). Here, \(a = \sin x\) and \(b = \cos x\).Thus,\[(\sin x+\cos x)^{2} = (\sin x)^2 + 2(\sin x)(\cos x) + (\cos x)^2\] This simplifies to:\[\sin^2 x + 2\sin x \cos x + \cos^2 x\]

Step 2: Simplify using Trigonometric Identity

From Step 1, we have \(\sin^2 x + \cos^2 x\) as part of the expression. We can simplify this using the Pythagorean identity:\[\sin^2 x + \cos^2 x = 1\]Substitute this into our equation from Step 1:\[1 + 2\sin x \cos x\]

Step 3: Use Double Angle Formula

The equation from Step 2 is \(1 + 2\sin x \cos x\). Now, let's use the double angle formula for sine: \[\sin 2x = 2\sin x \cos x\]This allows us to substitute the term \(2\sin x \cos x\) with \(\sin 2x\):\[1 + \sin 2x\]

Step 4: Conclude the Proof

At this point, we have shown that the left side \((\sin x + \cos x)^2\) simplifies to \(1 + \sin 2x\), which matches the right side of the identity we wanted to prove. Thus, we have:\[(\sin x + \cos x)^2 = 1 + \sin 2x\]This confirms that the identity is correct.

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry named after the Pythagorean theorem, which relates the sides of a right triangle. In its simplest form, it states that for any angle \(x\): \[sin^2 x + cos^2 x = 1.\]This identity is derived from the unit circle, where the radius is 1, making the square of the sine and cosine of an angle equal to 1. The Pythagorean Identity is frequently used to simplify trigonometric expressions such as \(\sin^2 x + \cos^2 x\) by replacing them with 1.

• It is an essential tool in both proving and simplifying trigonometric identities.
• This identity helps transition from one form of an equation to another seamlessly.

In our original exercise, after expanding \((\sin x + \cos x)^2\), we applied the Pythagorean Identity to simplify \(\sin^2 x + \cos^2 x\) to 1, streamlining the expression to help prove the identity.

Double Angle Formula

The Double Angle Formula is a set of trigonometric identities that express trigonometric functions of double angles \(2x\) in terms of \(x\). One of the most critical forms is for sine:\[\sin 2x = 2\sin x \cos x.\]This formula is helpful when working with problems where the expression involves products of \(\sin x\) and \(\cos x\). It allows conversion into a single sine term of a double angle measure.

• Simplifies mathematical expressions by reducing terms like \(2\sin x \cos x\).
• Useful while solving equations and proving identities where double angles occur.

In the proof of our identity, we substituted \(2\sin x \cos x\) from the expression in step 2 with \(\sin 2x\) using the Double Angle Formula, further bridging the gap between the expanded binomial and the right side of the asserted identity.

Binomial Expansion

Binomial Expansion is a powerful algebraic tool used to expand expressions raised to a power, represented commonly by the formula \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k.\]For the square of a binomial, this simplifies to \[(a + b)^2 = a^2 + 2ab + b^2.\]This method is particularly handy in breaking down expressions, making them easier to simplify or work with in further calculations.

• Allows easy manipulation of terms to apply other identities.
• Often the first step in many trigonometric proofs or expansions.

In our exercise, we expanded \((\sin x + \cos x)^2\) using the binomial expansion into \(\sin^2 x + 2\sin x \cos x + \cos^2 x\), setting the stage for the application of trigonometric identities and formulas to prove the original identity.