Verify The Identity:${ \frac{1+\cos \theta}{\sin \theta}+\frac{\sin \theta}{1+\cos \theta}=\frac{2}{\sin \theta} }$

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Introduction

In mathematics, trigonometric identities play a crucial role in solving various problems and equations. These identities help us simplify complex expressions and make calculations more manageable. In this article, we will focus on verifying a specific trigonometric identity involving sine and cosine functions. The given identity is:

1+cos⁡θsin⁡θ+sin⁡θ1+cos⁡θ=2sin⁡θ\frac{1+\cos \theta}{\sin \theta}+\frac{\sin \theta}{1+\cos \theta}=\frac{2}{\sin \theta}

Our goal is to prove that this identity holds true for all values of θ.

Step 1: Simplify the Left-Hand Side

To verify the identity, we will start by simplifying the left-hand side of the equation. We can do this by finding a common denominator for the two fractions.

1+cos⁡θsin⁡θ+sin⁡θ1+cos⁡θ=(1+cos⁡θ)2+sin⁡2θsin⁡θ(1+cos⁡θ)\frac{1+\cos \theta}{\sin \theta}+\frac{\sin \theta}{1+\cos \theta}=\frac{(1+\cos \theta)^2+\sin^2 \theta}{\sin \theta(1+\cos \theta)}

Now, we can expand the numerator using the formula (a+b)2=a2+2ab+b2(a+b)^2=a^2+2ab+b^2.

(1+cos⁡θ)2+sin⁡2θsin⁡θ(1+cos⁡θ)=1+2cos⁡θ+cos⁡2θ+sin⁡2θsin⁡θ(1+cos⁡θ)\frac{(1+\cos \theta)^2+\sin^2 \theta}{\sin \theta(1+\cos \theta)}=\frac{1+2\cos \theta+\cos^2 \theta+\sin^2 \theta}{\sin \theta(1+\cos \theta)}

Using the trigonometric identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta+\cos^2 \theta=1, we can simplify the numerator further.

1+2cos⁡θ+cos⁡2θ+sin⁡2θsin⁡θ(1+cos⁡θ)=1+2cos⁡θ+1sin⁡θ(1+cos⁡θ)\frac{1+2\cos \theta+\cos^2 \theta+\sin^2 \theta}{\sin \theta(1+\cos \theta)}=\frac{1+2\cos \theta+1}{\sin \theta(1+\cos \theta)}

Simplifying the numerator, we get:

1+2cos⁡θ+1sin⁡θ(1+cos⁡θ)=2+2cos⁡θsin⁡θ(1+cos⁡θ)\frac{1+2\cos \theta+1}{\sin \theta(1+\cos \theta)}=\frac{2+2\cos \theta}{\sin \theta(1+\cos \theta)}

Step 2: Factor Out the Common Term

We can factor out the common term 22 from the numerator.

2+2cos⁡θsin⁡θ(1+cos⁡θ)=2(1+cos⁡θ)sin⁡θ(1+cos⁡θ)\frac{2+2\cos \theta}{\sin \theta(1+\cos \theta)}=\frac{2(1+\cos \theta)}{\sin \theta(1+\cos \theta)}

Now, we can cancel out the common term (1+cos⁡θ)(1+\cos \theta) from the numerator and denominator.

2(1+cos⁡θ)sin⁡θ(1+cos⁡θ)=2sin⁡θ\frac{2(1+\cos \theta)}{\sin \theta(1+\cos \theta)}=\frac{2}{\sin \theta}

Conclusion

We have successfully verified the given trigonometric identity. By simplifying the left-hand side of the equation and factoring out the common term, we were able to show that the identity holds true for all values of θ.

The Importance of Trigonometric Identities

Trigonometric identities are essential in mathematics and have numerous applications in various fields, including physics, engineering, and computer science. They help us simplify complex expressions and make calculations more manageable. In this article, we have demonstrated the importance of trigonometric identities by verifying a specific identity involving sine and cosine functions.

Common Trigonometric Identities

Here are some common trigonometric identities that you may find useful:

  • sin⁡2θ+cos⁡2θ=1\sin^2 \theta+\cos^2 \theta=1
  • tan⁡θ=sin⁡θcos⁡θ\tan \theta=\frac{\sin \theta}{\cos \theta}
  • cot⁡θ=cos⁡θsin⁡θ\cot \theta=\frac{\cos \theta}{\sin \theta}
  • sec⁡θ=1cos⁡θ\sec \theta=\frac{1}{\cos \theta}
  • csc⁡θ=1sin⁡θ\csc \theta=\frac{1}{\sin \theta}

Tips for Verifying Trigonometric Identities

When verifying trigonometric identities, it's essential to follow these tips:

  • Start by simplifying the left-hand side of the equation.
  • Use trigonometric identities to simplify the expression.
  • Factor out common terms to cancel out common factors.
  • Check your work by plugging in values of θ to verify the identity.

Conclusion

Q: What is the purpose of verifying trigonometric identities?

A: Verifying trigonometric identities is essential in mathematics as it helps us simplify complex expressions and make calculations more manageable. By verifying identities, we can ensure that our calculations are accurate and reliable.

Q: How do I start verifying a trigonometric identity?

A: To start verifying a trigonometric identity, begin by simplifying the left-hand side of the equation. Use trigonometric identities to simplify the expression and factor out common terms to cancel out common factors.

Q: What are some common trigonometric identities that I should know?

A: Here are some common trigonometric identities that you may find useful:

  • sin⁡2θ+cos⁡2θ=1\sin^2 \theta+\cos^2 \theta=1
  • tan⁡θ=sin⁡θcos⁡θ\tan \theta=\frac{\sin \theta}{\cos \theta}
  • cot⁡θ=cos⁡θsin⁡θ\cot \theta=\frac{\cos \theta}{\sin \theta}
  • sec⁡θ=1cos⁡θ\sec \theta=\frac{1}{\cos \theta}
  • csc⁡θ=1sin⁡θ\csc \theta=\frac{1}{\sin \theta}

Q: How do I factor out common terms in a trigonometric expression?

A: To factor out common terms in a trigonometric expression, look for terms that have a common factor. For example, if you have the expression 2sin⁡θ+2cos⁡θ2\sin \theta+2\cos \theta, you can factor out the common term 22 to get 2(sin⁡θ+cos⁡θ)2(\sin \theta+\cos \theta).

Q: What is the difference between a trigonometric identity and a trigonometric equation?

A: A trigonometric identity is an equation that is true for all values of the variable, while a trigonometric equation is an equation that is true for specific values of the variable. For example, the equation sin⁡2θ+cos⁡2θ=1\sin^2 \theta+\cos^2 \theta=1 is a trigonometric identity, while the equation sin⁡θ=0\sin \theta=0 is a trigonometric equation.

Q: How do I check my work when verifying a trigonometric identity?

A: To check your work when verifying a trigonometric identity, plug in values of θ to verify that the identity holds true. For example, you can plug in θ=0, θ=π/2, and θ=π to verify that the identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta+\cos^2 \theta=1 holds true.

Q: What are some common mistakes to avoid when verifying trigonometric identities?

A: Here are some common mistakes to avoid when verifying trigonometric identities:

  • Not simplifying the left-hand side of the equation
  • Not factoring out common terms
  • Not checking your work by plugging in values of θ
  • Not using trigonometric identities to simplify the expression

Q: How can I practice verifying trigonometric identities?

A: You can practice verifying trigonometric identities by working through examples and exercises in your textbook or online resources. You can also try verifying identities on your own by using online tools or software.

Q: What are some real-world applications of trigonometric identities?

A: Trigonometric identities have numerous real-world applications in fields such as physics, engineering, and computer science. For example, trigonometric identities are used to describe the motion of objects in physics, to design electronic circuits in engineering, and to create 3D graphics in computer science.

Conclusion

Verifying trigonometric identities is an essential skill in mathematics that has numerous real-world applications. By following the steps outlined in this article, you can verify identities involving sine and cosine functions. Remember to simplify the left-hand side of the equation, factor out common terms, and check your work by plugging in values of θ. With practice, you will become proficient in verifying trigonometric identities and be able to apply them to solve complex problems.