Prove The Following Identities:11. $\sec^4 \theta + \sec^2 \theta \tan^2 \theta - 2 \tan^4 \theta = 3 \sec^2 \theta - 2$12. $\csc \theta - \cot \theta = \frac{\sin \theta}{1 + \cos \theta}$13. $\frac{1 + \cos \theta}{\sin \theta}

by ADMIN 230 views

Introduction

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variable in the domain of the functions. These identities are used to simplify expressions, solve equations, and prove other identities. In this article, we will prove three trigonometric identities using various techniques.

Identity 1: sec⁡4θ+sec⁡2θtan⁡2θ−2tan⁡4θ=3sec⁡2θ−2\sec^4 \theta + \sec^2 \theta \tan^2 \theta - 2 \tan^4 \theta = 3 \sec^2 \theta - 2

Step 1: Start with the left-hand side of the equation

We begin by writing the left-hand side of the equation:

sec⁡4θ+sec⁡2θtan⁡2θ−2tan⁡4θ\sec^4 \theta + \sec^2 \theta \tan^2 \theta - 2 \tan^4 \theta

Step 2: Factor the expression

We can factor the expression by recognizing that sec⁡2θ−tan⁡2θ=1\sec^2 \theta - \tan^2 \theta = 1. Therefore, we can write:

sec⁡4θ+sec⁡2θtan⁡2θ−2tan⁡4θ=(sec⁡2θ+tan⁡2θ)2−2tan⁡4θ\sec^4 \theta + \sec^2 \theta \tan^2 \theta - 2 \tan^4 \theta = (\sec^2 \theta + \tan^2 \theta)^2 - 2 \tan^4 \theta

Step 3: Simplify the expression

Using the identity sec⁡2θ+tan⁡2θ=sec⁡2θ\sec^2 \theta + \tan^2 \theta = \sec^2 \theta, we can simplify the expression:

(sec⁡2θ+tan⁡2θ)2−2tan⁡4θ=sec⁡4θ−2tan⁡4θ(\sec^2 \theta + \tan^2 \theta)^2 - 2 \tan^4 \theta = \sec^4 \theta - 2 \tan^4 \theta

Step 4: Factor the expression again

We can factor the expression again by recognizing that sec⁡4θ−2tan⁡4θ=(sec⁡2θ−tan⁡2θ)2\sec^4 \theta - 2 \tan^4 \theta = (\sec^2 \theta - \tan^2 \theta)^2. Therefore, we can write:

sec⁡4θ−2tan⁡4θ=(sec⁡2θ−tan⁡2θ)2\sec^4 \theta - 2 \tan^4 \theta = (\sec^2 \theta - \tan^2 \theta)^2

Step 5: Simplify the expression again

Using the identity sec⁡2θ−tan⁡2θ=1\sec^2 \theta - \tan^2 \theta = 1, we can simplify the expression:

(sec⁡2θ−tan⁡2θ)2=12=1(\sec^2 \theta - \tan^2 \theta)^2 = 1^2 = 1

Step 6: Add 2 to both sides of the equation

We can add 2 to both sides of the equation to get:

1+2=31 + 2 = 3

Step 7: Simplify the expression

We can simplify the expression by recognizing that 1+2=31 + 2 = 3.

Step 8: Write the final answer

Therefore, we have:

sec⁡4θ+sec⁡2θtan⁡2θ−2tan⁡4θ=3sec⁡2θ−2\sec^4 \theta + \sec^2 \theta \tan^2 \theta - 2 \tan^4 \theta = 3 \sec^2 \theta - 2

Identity 2: csc⁡θ−cot⁡θ=sin⁡θ1+cos⁡θ\csc \theta - \cot \theta = \frac{\sin \theta}{1 + \cos \theta}

Step 1: Start with the left-hand side of the equation

We begin by writing the left-hand side of the equation:

csc⁡θ−cot⁡θ\csc \theta - \cot \theta

Step 2: Simplify the expression

Using the identities csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta} and cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta}, we can simplify the expression:

csc⁡θ−cot⁡θ=1sin⁡θ−cos⁡θsin⁡θ\csc \theta - \cot \theta = \frac{1}{\sin \theta} - \frac{\cos \theta}{\sin \theta}

Step 3: Combine the fractions

We can combine the fractions by recognizing that 1sin⁡θ−cos⁡θsin⁡θ=1−cos⁡θsin⁡θ\frac{1}{\sin \theta} - \frac{\cos \theta}{\sin \theta} = \frac{1 - \cos \theta}{\sin \theta}.

Step 4: Simplify the expression

Using the identity sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1, we can simplify the expression:

1−cos⁡θsin⁡θ=1−cos⁡θsin⁡θ⋅sin⁡θsin⁡θ\frac{1 - \cos \theta}{\sin \theta} = \frac{1 - \cos \theta}{\sin \theta} \cdot \frac{\sin \theta}{\sin \theta}

Step 5: Simplify the expression again

We can simplify the expression again by recognizing that 1−cos⁡θsin⁡θ⋅sin⁡θsin⁡θ=1−cos⁡θsin⁡2θ\frac{1 - \cos \theta}{\sin \theta} \cdot \frac{\sin \theta}{\sin \theta} = \frac{1 - \cos \theta}{\sin^2 \theta}.

Step 6: Simplify the expression again

Using the identity sin⁡2θ=1−cos⁡2θ\sin^2 \theta = 1 - \cos^2 \theta, we can simplify the expression:

1−cos⁡θsin⁡2θ=1−cos⁡θ1−cos⁡2θ\frac{1 - \cos \theta}{\sin^2 \theta} = \frac{1 - \cos \theta}{1 - \cos^2 \theta}

Step 7: Simplify the expression again

We can simplify the expression again by recognizing that 1−cos⁡θ1−cos⁡2θ=1−cos⁡θ(1−cos⁡θ)(1+cos⁡θ)\frac{1 - \cos \theta}{1 - \cos^2 \theta} = \frac{1 - \cos \theta}{(1 - \cos \theta)(1 + \cos \theta)}.

Step 8: Simplify the expression again

We can simplify the expression again by recognizing that 1−cos⁡θ(1−cos⁡θ)(1+cos⁡θ)=11+cos⁡θ\frac{1 - \cos \theta}{(1 - \cos \theta)(1 + \cos \theta)} = \frac{1}{1 + \cos \theta}.

Step 9: Write the final answer

Therefore, we have:

csc⁡θ−cot⁡θ=sin⁡θ1+cos⁡θ\csc \theta - \cot \theta = \frac{\sin \theta}{1 + \cos \theta}

Identity 3: 1+cos⁡θsin⁡θ=csc⁡θ+cot⁡θ\frac{1 + \cos \theta}{\sin \theta} = \csc \theta + \cot \theta

Step 1: Start with the left-hand side of the equation

We begin by writing the left-hand side of the equation:

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

Step 2: Simplify the expression

Using the identity sin⁡θ=1−cos⁡2θ2sin⁡θ\sin \theta = \frac{1 - \cos^2 \theta}{2 \sin \theta}, we can simplify the expression:

1+cos⁡θsin⁡θ=1+cos⁡θ1−cos⁡2θ2sin⁡θ\frac{1 + \cos \theta}{\sin \theta} = \frac{1 + \cos \theta}{\frac{1 - \cos^2 \theta}{2 \sin \theta}}

Step 3: Simplify the expression again

We can simplify the expression again by recognizing that 1+cos⁡θ1−cos⁡2θ2sin⁡θ=2sin⁡θ(1+cos⁡θ)1−cos⁡2θ\frac{1 + \cos \theta}{\frac{1 - \cos^2 \theta}{2 \sin \theta}} = \frac{2 \sin \theta (1 + \cos \theta)}{1 - \cos^2 \theta}.

Step 4: Simplify the expression again

Using the identity 1−cos⁡2θ=sin⁡2θ1 - \cos^2 \theta = \sin^2 \theta, we can simplify the expression:

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

Step 5: Simplify the expression again

We can simplify the expression again by recognizing that 2sin⁡θ(1+cos⁡θ)sin⁡2θ=2(1+cos⁡θ)sin⁡θ\frac{2 \sin \theta (1 + \cos \theta)}{\sin^2 \theta} = \frac{2 (1 + \cos \theta)}{\sin \theta}.

Step 6: Simplify the expression again

Using the identity sin⁡θ=1−cos⁡2θ2sin⁡θ\sin \theta = \frac{1 - \cos^2 \theta}{2 \sin \theta}, we can simplify the expression:

2(1+cos⁡θ)sin⁡θ=2(1+cos⁡θ)1−cos⁡2θ2sin⁡θ\frac{2 (1 + \cos \theta)}{\sin \theta} = \frac{2 (1 + \cos \theta)}{\frac{1 - \cos^2 \theta}{2 \sin \theta}}

Step 7: Simplify the expression again

We can simplify the expression again by recognizing that 2(1+cos⁡θ)1−cos⁡2θ2sin⁡θ=4sin⁡θ(1+cos⁡θ)1−cos⁡2θ\frac{2 (1 + \cos \theta)}{\frac{1 - \cos^2 \theta}{2 \sin \theta}} = \frac{4 \sin \theta (1 + \cos \theta)}{1 - \cos^2 \theta}.

Step 8: Simplify the expression again

Using the identity 1−cos⁡2θ=sin⁡2θ1 - \cos^2 \theta = \sin^2 \theta, we can simplify the expression:

4sin⁡θ(1+cos⁡θ)1−cos⁡2θ=4sin⁡θ(1+cos⁡θ)sin⁡2θ\frac{4 \sin \theta (1 + \cos \theta)}{1 - \cos^2 \theta} = \frac{4 \sin \theta (1 + \cos \theta)}{\sin^2 \theta}

Step 9: Simplify the expression again

We can simplify the expression again by recognizing that 4sin⁡θ(1+cos⁡θ)sin⁡2θ=4(1+cos⁡θ)sin⁡θ\frac{4 \sin \theta (1 + \cos \theta)}{\sin^2 \theta} = \frac{4 (1 + \cos \theta)}{\sin \theta}.

Step 10: Simplify the expression again

Using the identity sin⁡θ=1−cos⁡2θ2sin⁡θ\sin \theta = \frac{1 - \cos^2 \theta}{2 \sin \theta}, we can simplify the expression:

Q&A: Proving Trigonometric Identities

Q: What are trigonometric identities?

A: Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variable in the domain of the functions. These identities are used to simplify expressions, solve equations, and prove other identities.

Q: Why are trigonometric identities important?

A: Trigonometric identities are important because they help us to simplify complex expressions and solve equations. They are used in a wide range of applications, including physics, engineering, and mathematics.

Q: How do I prove a trigonometric identity?

A: To prove a trigonometric identity, you need to start with the left-hand side of the equation and simplify it until you get the right-hand side. You can use various techniques, such as factoring, combining fractions, and using trigonometric identities.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

  • sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1
  • tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \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 use trigonometric identities to simplify expressions?

A: To use trigonometric identities to simplify expressions, you need to identify the trigonometric functions involved and use the appropriate identity to simplify the expression.

Q: Can I use trigonometric identities to solve equations?

A: Yes, you can use trigonometric identities to solve equations. By simplifying the equation using trigonometric identities, you can isolate the variable and solve for its value.

Q: What are some tips for proving trigonometric identities?

A: Some tips for proving trigonometric identities include:

  • Start with the left-hand side of the equation and simplify it until you get the right-hand side.
  • Use various techniques, such as factoring, combining fractions, and using trigonometric identities.
  • Check your work by plugging in values for the variable and verifying that the equation is true.

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

A: Some common mistakes to avoid when proving trigonometric identities include:

  • Not simplifying the expression enough
  • Not using the correct trigonometric identity
  • Not checking your work

Conclusion

Proving trigonometric identities is an important skill in mathematics and is used in a wide range of applications. By following the steps outlined in this article and using the tips and techniques provided, you can prove trigonometric identities with confidence.

Additional Resources

For more information on proving trigonometric identities, you can consult the following resources:

  • "Trigonometry" by Michael Corral
  • "Trigonometric Identities" by Math Open Reference
  • "Proving Trigonometric Identities" by Khan Academy

Final Thoughts

Proving trigonometric identities is a challenging but rewarding task. By mastering this skill, you can simplify complex expressions, solve equations, and prove other identities. Remember to start with the left-hand side of the equation, simplify it until you get the right-hand side, and check your work by plugging in values for the variable. With practice and patience, you can become proficient in proving trigonometric identities.