Question

Help me verify this identity.Follow the structure from the notes by identify the properties used in each step.

Answer 1

Given that you have to verify that:

`(1)/(1-sinx)+(1-sinx)/(cos^2x)=2sec^2x`

You need to follow these steps, in order to solve the exercise:

1. You need to use this Pythagorean Identify to rewrite the denominator of the second fraction:

`cos^2x+sin^2x=1`

If you solve for:

`cos^2x`

You get:

`cos^2x=1-sin^2x`

Then, you can rewrite the expression on the right side in this form:

`(1)/(1-sinx)+(1-sinx)/(1-sin^2x)=2sec^2x`

2. By definition:

`(a-b)(a+b)=a^2-b^2`

Then, you can rewrite the denominator of the second fraction in this form:

`(1)/(1-sinx)+(1-sinx)/((1-sinx)(1+sinx))=2sec^2x`

3. Now you can cancel common terms:

`(1)/(1-sinx)+(1)/(1+sinx)=2sec^2x`

4. Add the fractions using this formula for adding fractions with different denominators:

`(a)/(b)+(c)/(d)=(ad+bc)/(ad)`

Then, you get:

`((1)(1+sinx)+(1)(1-sinx))/((1-sinx)(1+sinx))=2sec^2x`
`(1+sinx+1-sinx)/((1-sinx)(1+sinx))=2sec^2x`
`(2)/(1-sin^2x)=2sec^2x`

5. You already know that:

`cos^2x=1-sin^2x`

Then, you can rewrite the denominator as:

`(2)/(cos^2x)=2sec^2x`

6. Remember this Reciprocal Identify:

`(1)/(cosx)=secx`

Therefore, you can determine:

`2sec^2x=2sec^2x`

Hence, the answer is: