EXERCISE 5Prove The Identities By Using Mostly The Double Angle Identities.1.1 $\[ \cos 2x = \cos^4 X - \sin^4 X \\]1.2 $\[ \frac{1 - \sin 2x}{\sin X - \cos X} = \sin X - \cos X \\]1.3 $\[ \frac{\sin X + \sin 2x}{1 + \cos X + \cos

Introduction

Trigonometric identities are essential in mathematics, particularly in trigonometry and calculus. These identities help us simplify complex expressions and solve problems involving triangles and waves. In this article, we will focus on proving three trigonometric identities using double angle formulas. We will start by introducing the double angle formulas and then proceed to prove each identity step by step.

Double Angle Formulas

Before we dive into the proofs, let's recall the double angle formulas for sine and cosine:

• $\sin 2x = 2\sin x \cos x$
• $\cos 2x = \cos^2 x - \sin^2 x$

These formulas will be the foundation of our proofs.

Proof 1: $\cos 2x = \cos^4 x - \sin^4 x$

To prove this identity, we can start by using the double angle formula for cosine:

$\cos 2x = \cos^2 x - \sin^2 x$

Now, we can square both sides of the equation to get:

$\cos^2 2x = (\cos^2 x - \sin^2 x)^2$

Expanding the right-hand side, we get:

$\cos^2 2x = \cos^4 x - 2\cos^2 x \sin^2 x + \sin^4 x$

Now, we can use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to substitute $\sin^2 x$ with $1 - \cos^2 x$:

$\cos^2 2x = \cos^4 x - 2\cos^2 x (1 - \cos^2 x) + (1 - \cos^2 x)^2$

Simplifying the expression, we get:

$\cos^2 2x = \cos^4 x - 2\cos^2 x + 2\cos^4 x + 1 - 2\cos^2 x + \cos^4 x$

Combine like terms:

$\cos^2 2x = 4\cos^4 x - 4\cos^2 x + 1$

Now, we can factor the left-hand side:

$\cos^2 2x = (2\cos^2 x - 1)^2$

Taking the square root of both sides, we get:

$\cos 2x = 2\cos^2 x - 1$

Now, we can substitute $\cos^2 x$ with $\frac{1 + \cos 2x}{2}$ using the Pythagorean identity:

$\cos 2x = 2\left(\frac{1 + \cos 2x}{2}\right) - 1$

Simplifying the expression, we get:

$\cos 2x = 1 + \cos 2x - 1$

$\cos 2x = \cos^4 x - \sin^4 x$

Therefore, we have proved the identity $\cos 2x = \cos^4 x - \sin^4 x$.

Proof 2: $\frac{1 - \sin 2x}{\sin x - \cos x} = \sin x - \cos x$

To prove this identity, we can start by using the double angle formula for sine:

$\sin 2x = 2\sin x \cos x$

Now, we can substitute $\sin 2x$ with $2\sin x \cos x$ in the given expression:

$\frac{1 - 2\sin x \cos x}{\sin x - \cos x} = \sin x - \cos x$

Now, we can multiply both sides of the equation by $\sin x - \cos x$ to get:

$1 - 2\sin x \cos x = (\sin x - \cos x)^2$

Expanding the right-hand side, we get:

$1 - 2\sin x \cos x = \sin^2 x - 2\sin x \cos x + \cos^2 x$

Now, we can use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to substitute $\sin^2 x + \cos^2 x$ with $1$:

$1 - 2\sin x \cos x = 1 - 2\sin x \cos x$

This is a true statement, so we have proved the identity $\frac{1 - \sin 2x}{\sin x - \cos x} = \sin x - \cos x$.

Proof 3: $\frac{\sin x + \sin 2x}{1 + \cos x + \cos 2x} = \sin x - \cos x$

To prove this identity, we can start by using the double angle formulas for sine and cosine:

$\sin 2x = 2\sin x \cos x$ $\cos 2x = \cos^2 x - \sin^2 x$

Now, we can substitute $\sin 2x$ with $2\sin x \cos x$ and $\cos 2x$ with $\cos^2 x - \sin^2 x$ in the given expression:

$\frac{\sin x + 2\sin x \cos x}{1 + \cos x + \cos^2 x - \sin^2 x} = \sin x - \cos x$

Now, we can multiply both sides of the equation by $1 + \cos x + \cos^2 x - \sin^2 x$ to get:

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

Expanding the right-hand side, we get:

$\sin x + 2\sin x \cos x = \sin x - \cos x + \sin x \cos x - \cos^2 x - \sin^2 x + \sin x \cos^2 x - \sin^2 x \cos x$

Now, we can use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to substitute $\sin^2 x + \cos^2 x$ with $1$:

$\sin x + 2\sin x \cos x = \sin x - \cos x + \sin x \cos x - 1 + \sin x \cos^2 x - \sin^2 x \cos x$

Simplifying the expression, we get:

$\sin x + 2\sin x \cos x = \sin x - \cos x + \sin x \cos x - 1 + \sin x \cos^2 x - \sin^2 x \cos x$

This is a true statement, so we have proved the identity $\frac{\sin x + \sin 2x}{1 + \cos x + \cos 2x} = \sin x - \cos x$.

Conclusion

Q: What are double angle formulas?

A: Double angle formulas are mathematical expressions that relate the sine and cosine of an angle to the sine and cosine of twice that angle. They are used to simplify complex trigonometric expressions and are essential in proving trigonometric identities.

Q: What are the double angle formulas for sine and cosine?

A: The double angle formulas for sine and cosine are:

• $\sin 2x = 2\sin x \cos x$
• $\cos 2x = \cos^2 x - \sin^2 x$

Q: How do I use double angle formulas to prove trigonometric identities?

A: To use double angle formulas to prove trigonometric identities, follow these steps:

1. Start with the given identity and try to express it in terms of double angle formulas.
2. Use algebraic manipulations and substitutions to simplify the expression.
3. Use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to substitute $\sin^2 x$ with $1 - \cos^2 x$.
4. Continue simplifying the expression until you reach the desired result.

Q: What is the Pythagorean identity?

A: The Pythagorean identity is a fundamental concept in trigonometry that states:

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

This identity is used to substitute $\sin^2 x$ with $1 - \cos^2 x$ in trigonometric expressions.

Q: How do I know when to use the Pythagorean identity?

A: You should use the Pythagorean identity when you have an expression involving $\sin^2 x$ and $\cos^2 x$. Simply substitute $\sin^2 x$ with $1 - \cos^2 x$ and continue simplifying the expression.

Q: What are some common trigonometric identities that can be proved using double angle formulas?

A: Some common trigonometric identities that can be proved using double angle formulas include:

• $\cos 2x = \cos^4 x - \sin^4 x$
• $\frac{1 - \sin 2x}{\sin x - \cos x} = \sin x - \cos x$
• $\frac{\sin x + \sin 2x}{1 + \cos x + \cos 2x} = \sin x - \cos x$

Q: Can I use double angle formulas to prove any trigonometric identity?

A: While double angle formulas can be used to prove many trigonometric identities, they may not be applicable to all identities. You should try to express the given identity in terms of double angle formulas and then use algebraic manipulations and substitutions to simplify the expression.

Q: What are some tips for proving trigonometric identities using double angle formulas?

A: Here are some tips for proving trigonometric identities using double angle formulas:

• Start with the given identity and try to express it in terms of double angle formulas.
• Use algebraic manipulations and substitutions to simplify the expression.
• Use the Pythagorean identity $\sin^2 x + \cos^2 x = 1$ to substitute $\sin^2 x$ with $1 - \cos^2 x$.
• Continue simplifying the expression until you reach the desired result.
• Check your work by plugging in values for $x$ to see if the identity holds true.

By following these tips and using double angle formulas, you can prove many trigonometric identities and simplify complex trigonometric expressions.