Simplify The Expression:$ (\cos \theta - \sin \theta)^2 = 1 - \sin 2\theta $

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Simplify the Expression: (cos⁡θ−sin⁡θ)2=1−sin⁡2θ(\cos \theta - \sin \theta)^2 = 1 - \sin 2\theta

In this article, we will explore the simplification of a trigonometric expression involving the cosine and sine functions. The given expression is (cos⁡θ−sin⁡θ)2=1−sin⁡2θ(\cos \theta - \sin \theta)^2 = 1 - \sin 2\theta. We will use various trigonometric identities and formulas to simplify this expression and arrive at the given result.

The given expression involves the square of a binomial expression, which is (cos⁡θ−sin⁡θ)2(\cos \theta - \sin \theta)^2. To simplify this expression, we will use the formula for expanding a binomial square, which is (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2. In this case, a=cos⁡θa = \cos \theta and b=sin⁡θb = \sin \theta.

Using the formula for expanding a binomial square, we can expand the given expression as follows:

(cos⁡θ−sin⁡θ)2=(cos⁡θ)2−2(cos⁡θ)(sin⁡θ)+(sin⁡θ)2(\cos \theta - \sin \theta)^2 = (\cos \theta)^2 - 2(\cos \theta)(\sin \theta) + (\sin \theta)^2

We can simplify the expanded expression by applying various trigonometric identities. The first identity we will use is the Pythagorean identity, which states that sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1. We can rewrite the expanded expression as follows:

(cos⁡θ)2−2(cos⁡θ)(sin⁡θ)+(sin⁡θ)2=cos⁡2θ−2sin⁡θcos⁡θ+sin⁡2θ(\cos \theta)^2 - 2(\cos \theta)(\sin \theta) + (\sin \theta)^2 = \cos^2 \theta - 2\sin \theta \cos \theta + \sin^2 \theta

We can simplify the expression further by using the Pythagorean identity. We can rewrite the expression as follows:

cos⁡2θ−2sin⁡θcos⁡θ+sin⁡2θ=(cos⁡2θ+sin⁡2θ)−2sin⁡θcos⁡θ\cos^2 \theta - 2\sin \theta \cos \theta + \sin^2 \theta = (\cos^2 \theta + \sin^2 \theta) - 2\sin \theta \cos \theta

Using the Pythagorean identity, we can simplify the expression as follows:

(cos⁡2θ+sin⁡2θ)−2sin⁡θcos⁡θ=1−2sin⁡θcos⁡θ(\cos^2 \theta + \sin^2 \theta) - 2\sin \theta \cos \theta = 1 - 2\sin \theta \cos \theta

We can simplify the expression further by using the double angle formula for sine, which states that sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin \theta \cos \theta. We can rewrite the expression as follows:

1−2sin⁡θcos⁡θ=1−sin⁡2θ1 - 2\sin \theta \cos \theta = 1 - \sin 2\theta

In this article, we have simplified the expression (cos⁡θ−sin⁡θ)2=1−sin⁡2θ(\cos \theta - \sin \theta)^2 = 1 - \sin 2\theta using various trigonometric identities and formulas. We have expanded the binomial square, applied the Pythagorean identity, and used the double angle formula for sine to arrive at the given result. This expression is a fundamental result in trigonometry and has many applications in various fields, including physics, engineering, and mathematics.

  • The expression (cos⁡θ−sin⁡θ)2=1−sin⁡2θ(\cos \theta - \sin \theta)^2 = 1 - \sin 2\theta can be simplified using various trigonometric identities and formulas.
  • The Pythagorean identity is a fundamental identity in trigonometry and is used to simplify expressions involving sine and cosine.
  • The double angle formula for sine is a useful formula in trigonometry and is used to simplify expressions involving sine and cosine.

For further reading on trigonometry, we recommend the following resources:

  • "Trigonometry" by Michael Corral
  • "Trigonometry for Dummies" by Mary Jane Sterling
  • "Trigonometry: A Unit Circle Approach" by Charles P. McKeague and Mark D. Turner
  • Binomial square: A mathematical expression of the form (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2.
  • Pythagorean identity: A fundamental identity in trigonometry that states sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1.
  • Double angle formula: A formula in trigonometry that states sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin \theta \cos \theta.
  • Corral, M. (2013). Trigonometry. Pearson Education.
  • Sterling, M. J. (2013). Trigonometry for Dummies. John Wiley & Sons.
  • McKeague, C. P., & Turner, M. D. (2013). Trigonometry: A Unit Circle Approach. Cengage Learning.
    Simplify the Expression: (cos⁡θ−sin⁡θ)2=1−sin⁡2θ(\cos \theta - \sin \theta)^2 = 1 - \sin 2\theta - Q&A

In our previous article, we explored the simplification of a trigonometric expression involving the cosine and sine functions. The given expression is (cos⁡θ−sin⁡θ)2=1−sin⁡2θ(\cos \theta - \sin \theta)^2 = 1 - \sin 2\theta. We used various trigonometric identities and formulas to simplify this expression and arrive at the given result. In this article, we will answer some frequently asked questions related to this expression.

Q: What is the significance of the expression (cos⁡θ−sin⁡θ)2=1−sin⁡2θ(\cos \theta - \sin \theta)^2 = 1 - \sin 2\theta?

A: The expression (cos⁡θ−sin⁡θ)2=1−sin⁡2θ(\cos \theta - \sin \theta)^2 = 1 - \sin 2\theta is a fundamental result in trigonometry and has many applications in various fields, including physics, engineering, and mathematics. It is used to simplify expressions involving sine and cosine and is a useful tool in solving trigonometric problems.

Q: How can I simplify the expression (cos⁡θ−sin⁡θ)2=1−sin⁡2θ(\cos \theta - \sin \theta)^2 = 1 - \sin 2\theta?

A: To simplify the expression, you can use the formula for expanding a binomial square, which is (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2. In this case, a=cos⁡θa = \cos \theta and b=sin⁡θb = \sin \theta. You can then apply the Pythagorean identity and the double angle formula for sine to simplify the expression.

Q: What is the Pythagorean identity and how is it used in simplifying the expression?

A: The Pythagorean identity is a fundamental identity in trigonometry that states sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1. It is used to simplify expressions involving sine and cosine. In the expression (cos⁡θ−sin⁡θ)2=1−sin⁡2θ(\cos \theta - \sin \theta)^2 = 1 - \sin 2\theta, the Pythagorean identity is used to simplify the expression by rewriting it as (cos⁡2θ+sin⁡2θ)−2sin⁡θcos⁡θ=1−2sin⁡θcos⁡θ(\cos^2 \theta + \sin^2 \theta) - 2\sin \theta \cos \theta = 1 - 2\sin \theta \cos \theta.

Q: What is the double angle formula and how is it used in simplifying the expression?

A: The double angle formula is a formula in trigonometry that states sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin \theta \cos \theta. It is used to simplify expressions involving sine and cosine. In the expression (cos⁡θ−sin⁡θ)2=1−sin⁡2θ(\cos \theta - \sin \theta)^2 = 1 - \sin 2\theta, the double angle formula is used to simplify the expression by rewriting it as 1−2sin⁡θcos⁡θ=1−sin⁡2θ1 - 2\sin \theta \cos \theta = 1 - \sin 2\theta.

Q: Can I use this expression to solve trigonometric problems?

A: Yes, the expression (cos⁡θ−sin⁡θ)2=1−sin⁡2θ(\cos \theta - \sin \theta)^2 = 1 - \sin 2\theta can be used to solve trigonometric problems. It is a useful tool in simplifying expressions involving sine and cosine and can be used to solve problems involving trigonometric functions.

Q: Are there any other ways to simplify the expression (cos⁡θ−sin⁡θ)2=1−sin⁡2θ(\cos \theta - \sin \theta)^2 = 1 - \sin 2\theta?

A: Yes, there are other ways to simplify the expression (cos⁡θ−sin⁡θ)2=1−sin⁡2θ(\cos \theta - \sin \theta)^2 = 1 - \sin 2\theta. One way is to use the trigonometric identity cos⁡2θ=1−2sin⁡2θ\cos 2\theta = 1 - 2\sin^2 \theta to simplify the expression.

In this article, we have answered some frequently asked questions related to the expression (cos⁡θ−sin⁡θ)2=1−sin⁡2θ(\cos \theta - \sin \theta)^2 = 1 - \sin 2\theta. We have discussed the significance of the expression, how to simplify it, and how it can be used to solve trigonometric problems. We hope that this article has been helpful in understanding the expression and its applications.

  • The expression (cos⁡θ−sin⁡θ)2=1−sin⁡2θ(\cos \theta - \sin \theta)^2 = 1 - \sin 2\theta is a fundamental result in trigonometry and has many applications in various fields.
  • The Pythagorean identity and the double angle formula are used to simplify the expression.
  • The expression can be used to solve trigonometric problems involving sine and cosine.
  • There are other ways to simplify the expression, including using the trigonometric identity cos⁡2θ=1−2sin⁡2θ\cos 2\theta = 1 - 2\sin^2 \theta.

For further reading on trigonometry, we recommend the following resources:

  • "Trigonometry" by Michael Corral
  • "Trigonometry for Dummies" by Mary Jane Sterling
  • "Trigonometry: A Unit Circle Approach" by Charles P. McKeague and Mark D. Turner
  • Binomial square: A mathematical expression of the form (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2.
  • Pythagorean identity: A fundamental identity in trigonometry that states sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1.
  • Double angle formula: A formula in trigonometry that states sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin \theta \cos \theta.
  • Corral, M. (2013). Trigonometry. Pearson Education.
  • Sterling, M. J. (2013). Trigonometry for Dummies. John Wiley & Sons.
  • McKeague, C. P., & Turner, M. D. (2013). Trigonometry: A Unit Circle Approach. Cengage Learning.