Math Problem Statement

seca-1/seca+1=1-cosa/1+cosa

Solution

We are given the following trigonometric equation to solve:

1sec(A)1+sec(A)=1cos(A)1+cos(A)\frac{1 - \sec(A)}{1 + \sec(A)} = \frac{1 - \cos(A)}{1 + \cos(A)}

Step 1: Express sec(A)\sec(A) in terms of cos(A)\cos(A)

Since sec(A)=1cos(A)\sec(A) = \frac{1}{\cos(A)}, we can substitute this into the equation:

11cos(A)1+1cos(A)=1cos(A)1+cos(A)\frac{1 - \frac{1}{\cos(A)}}{1 + \frac{1}{\cos(A)}} = \frac{1 - \cos(A)}{1 + \cos(A)}

Step 2: Simplify the left-hand side

To simplify, multiply the numerator and denominator by cos(A)\cos(A):

cos(A)(11cos(A))cos(A)(1+1cos(A))=cos(A)1cos(A)+1\frac{\cos(A)(1 - \frac{1}{\cos(A)})}{\cos(A)(1 + \frac{1}{\cos(A)})} = \frac{\cos(A) - 1}{\cos(A) + 1}

Now the equation becomes:

cos(A)1cos(A)+1=1cos(A)1+cos(A)\frac{\cos(A) - 1}{\cos(A) + 1} = \frac{1 - \cos(A)}{1 + \cos(A)}

Step 3: Analyze the equation

The left-hand side is cos(A)1cos(A)+1\frac{\cos(A) - 1}{\cos(A) + 1}, while the right-hand side is 1cos(A)1+cos(A)\frac{1 - \cos(A)}{1 + \cos(A)}. Notice that:

1cos(A)=(cos(A)1)1 - \cos(A) = -( \cos(A) - 1)

So the right-hand side is the negative of the left-hand side:

1cos(A)1+cos(A)=cos(A)1cos(A)+1\frac{1 - \cos(A)}{1 + \cos(A)} = -\frac{\cos(A) - 1}{\cos(A) + 1}

Thus, the equation becomes:

cos(A)1cos(A)+1=cos(A)1cos(A)+1\frac{\cos(A) - 1}{\cos(A) + 1} = -\frac{\cos(A) - 1}{\cos(A) + 1}

Step 4: Solve for cos(A)\cos(A)

For this equation to hold, the only way for cos(A)1cos(A)+1=cos(A)1cos(A)+1\frac{\cos(A) - 1}{\cos(A) + 1} = -\frac{\cos(A) - 1}{\cos(A) + 1} is if:

cos(A)1=0\cos(A) - 1 = 0

This implies:

cos(A)=1\cos(A) = 1

Step 5: Conclusion

The solution to the equation is:

A=0+2nπfor any integernA = 0 + 2n\pi \quad \text{for any integer} \, n

Would you like more details on any step? Here are 5 related questions:

  1. What is the definition of sec(A)\sec(A) in trigonometry?
  2. How can trigonometric identities help simplify complex equations?
  3. What are the implications of cos(A)=1\cos(A) = 1 in trigonometry?
  4. How would the equation change if cos(A)1\cos(A) \neq 1?
  5. How do transformations affect trigonometric identities?

Tip: When simplifying trigonometric equations, always look for basic identities like sec(A)=1cos(A)\sec(A) = \frac{1}{\cos(A)} to help reduce complexity.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Trigonometry
Trigonometric Identities

Formulas

sec(A) = 1/cos(A)
cos(A)

Theorems

Trigonometric Identity Simplification
Cosine Value Theorem

Suitable Grade Level

Grades 10-12