# Please, check my work on Trig Identities

• solve
In summary, the conversation involved someone asking for their work to be checked for a homework assignment involving various mathematical equations. The expert summarizer provided a summary of their work, stating that parts (a) through (e) were correct and that they did not get a chance to look at part (f). They also advised the person that as long as they followed the rules of math and arrived at the final answer, their work is most likely correct.
solve

## Homework Statement

a) Show that sqrt{[1+tan^2x]/[1+cot^2x]}=tanx

b) Show that [cosx+sinx]/[cosx-sinx]=1+[2tanx]/[1- tanx]

c)Show that cotxcosx+tanxsinx=(cosecx+ secx)(1-sinxcosx)

d) Show that cosec^2x-cosecx=cot^2x/[1+sinx]

e) Show that sin^3x-cos^3x= (sinx-cosx)(1+sinxcosx)

f) Show that [cosx-1]/[secx+tanx]+[cosx+1]/[secx-tanx]=2(1+tanx)

## The Attempt at a Solution

a) sqrt{[1+tan^2x]/[1+cot^2x]}=tanx

=sqrt{[1+tan^2x]/[1+cot^2x]}

= sqrt{[sec^2x]/[cosec^2x]}

= secx/cosecx

= 1/cosx(sinx)

= tanx

b) [cosx+sinx]/[cosx-sinx]=1+[2tanx]/[1- tanx]

=1+[2tanx]/[1- tanx]

= [1-tanx+2tanx]/[1-tanx]

= [1+tanx]/[1-tanx]

= [1+ sinx/cosx]/[1-sinx/cosx]

= [(cosx+sinx)/cosx]/[(cosx-sinx)/cosx]

= [cosx+sinx]/[cosx-sinx)]

c) cotxcosx+tanxsinx=(cosecx+ secx)(1-sinxcosx)

=(cosecx+secx)(1-sinxcosx)

=(cosecx+secx)(sin^2x-sinxcosx+cos^2x)

= sinx+ [sin^2x/cosx]-cosx-sinx+[cos^2x/sinx]+cosx

= [sin^2x/cosx]+[cos^2x/sinx]

= [sinx/cosx](sinx)+[cosx/sinx](cosx)

= tanxsinx+cotxcosx

d) cosec^2x-cosecx=cot^2x/[1+sinx]

=cot^2x/[1+sinx]

= cot^2x/[1+sinx]*[(1-sinx)/(1-sinx)]

= [cot^2x-cot^2x(sinx)]/(1-sin^2x)

= [cot^2x-cot^2x(sinx)]/(cos^2x)

= [cot^2x]/[cos^2x]- [cot^2xsinx]/[cos^2x]

= [cos^2x]/[sin^2x](1/[cos^2x])- [cos^2x]/[sin^2x](1/[cos^2x])(sinx)

= 1/(sin^2x)-sinx/(sin^2x)

= cosec^2x-cosecx

e) sin^3x-cos^3x= (sinx-cosx)(1+sinxcosx)

= (sinx-cosx)(1+sinxcosx)

=( sinx-cosx)(sin^2x+sinxcosx+cos^2x)

= sin^3x-cos^3x

f) [cosx-1]/[secx+tanx]+[cosx+1]/[secx-tanx]=2(1+tanx)

= [cosx-1]/[secx+tanx]+[cosx+1]/[secx-tanx]

= [cosx-1]/[secx+tanx]+{[cosx+1]/[secx-tanx]*[(secx+tanx)/secx+tanx]}

= {(cosx-1)(secx-tanx)+(cosx+1)(secx+tanx)}/[sex^2x-tan^2x]

= 1-sinx-(1/cosx)+(sinx/cosx)+1+sinx+(1/cosx)+sinx/cosx

= 1+(sinx/cosx)+1+sinx/cosx

= 2+2(sinx/cosx)

= 2(1+tanx)

Thank You very much.

The work was a little difficult for me to read, even with the grouping symbols. But looks like (a) through (e) are correct. I didn't get a chance to look at (f).

(f) looks fine !

Thank you very much for checking my work, people. Much appreciated.

You shouldn't really need your work checked. As long as you're confident you didn't break any rules of math and arrive at the final answer it is most likely correct.

TimeToShine said:
You shouldn't really need your work checked. As long as you're confident you didn't break any rules of math
I think that was the point of the OP's post - he wasn't confident about his or her work. It takes quite a bit of practice to reach that point of confidence.
TimeToShine said:
and arrive at the final answer it is most likely correct.

## 1. What are trigonometric identities?

Trigonometric identities are equations that involve trigonometric functions (such as sine, cosine, and tangent) and are true for all values of the variables involved.

## 2. Why do we need to check our work on trig identities?

Checking our work on trig identities is important because it allows us to verify that our calculations are correct and that our solutions are valid. It also helps us to catch any mistakes or errors in our work.

## 3. How do we check our work on trig identities?

To check our work on trig identities, we can use a variety of methods such as substitution, simplification, or graphing. We can also use trigonometric identities to rewrite one side of the equation to match the other side, proving that they are equivalent.

## 4. What are some common mistakes to watch out for when working with trig identities?

Some common mistakes when working with trig identities include forgetting to use parentheses, making sign errors, and not simplifying properly. It is also important to check for extraneous solutions, which can occur when squaring or taking the square root of both sides of an equation.

## 5. How can I improve my skills in working with trig identities?

To improve your skills in working with trig identities, it is important to practice regularly and familiarize yourself with common identities and their applications. You can also try solving a variety of problems and checking your work to identify any areas that need improvement.

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