How Do You Simplify Trigonometric Expressions Using Basic Identities?

In summary: They should be ( ) instead, because you're not referring to the operation of a computer program, but to an algebraic expression.
  • #1
Jen23
12
0

Homework Statement


Express (1+cot^2 x) / (cot^2 x) in terms of sinx and/or cosx

Homework Equations


cot(x) = 1/tan(x)
sin^2(x) + cos^2(x) = 1

The Attempt at a Solution


I do not know if I am solving this problem correctly. Is there an easier route than the way I have solved it, if it is solved correctly?

= (1+cot^2x) / (cot^2x)
= 1+ [ (cos^2x) / (sin^2x) ] ÷ [ (cos^2x) / (sin^2x) ]
= 1 + [ (cos^2x) / (sin^2x) ] x [ (sin^2x / cos^2x) ]
= [ (sin^2x / sin^2x) + (cos^2x / sin^2x) ] x [ (sin^2x) / (cos^2x) ]
= [ (sin^2x + cos^2x) / (sin^2x) ] x [ (sin^2x )/ (cos^2x) ]
= [ 1 / sin^2x ] x [ sin^2x / cos^2x]
= sin^2x / (sin^2x)(cos^2x)
= 1 / cos^2x
 
Physics news on Phys.org
  • #2
Jen23 said:

Homework Statement


Express (1+cot^2 x) / (cot^2 x) in terms of sinx and/or cosx

Homework Equations


cot(x) = 1/tan(x)
sin^2(x) + cos^2(x) = 1

The Attempt at a Solution


I do not know if I am solving this problem correctly. Is there an easier route than the way I have solved it, if it is solved correctly?

= (1+cot^2x) / (cot^2x)
= 1+ [ (cos^2x) / (sin^2x) ] ÷ [ (cos^2x) / (sin^2x) ]
= 1 + [ (cos^2x) / (sin^2x) ] x [ (sin^2x / cos^2x) ]
= [ (sin^2x / sin^2x) + (cos^2x / sin^2x) ] x [ (sin^2x) / (cos^2x) ]
= [ (sin^2x + cos^2x) / (sin^2x) ] x [ (sin^2x )/ (cos^2x) ]
= [ 1 / sin^2x ] x [ sin^2x / cos^2x]
= sin^2x / (sin^2x)(cos^2x)
= 1 / cos^2x

Looks just fine to me.
 
  • #3
Jen23 said:

Homework Statement


Express (1+cot^2 x) / (cot^2 x) in terms of sinx and/or cosx

Homework Equations


cot(x) = 1/tan(x)
sin^2(x) + cos^2(x) = 1

The Attempt at a Solution


I do not know if I am solving this problem correctly. Is there an easier route than the way I have solved it, if it is solved correctly?

= (1+cot^2x) / (cot^2x)
= 1+ [ (cos^2x) / (sin^2x) ] ÷ [ (cos^2x) / (sin^2x) ]
= 1 + [ (cos^2x) / (sin^2x) ] x [ (sin^2x / cos^2x) ]
= [ (sin^2x / sin^2x) + (cos^2x / sin^2x) ] x [ (sin^2x) / (cos^2x) ]
= [ (sin^2x + cos^2x) / (sin^2x) ] x [ (sin^2x )/ (cos^2x) ]
= [ 1 / sin^2x ] x [ sin^2x / cos^2x]
= sin^2x / (sin^2x)(cos^2x)
= 1 / cos^2x

Why don't you use the fact that
$$
\frac{1 +\cot^2 x}{\cot^2 x} = \frac{1}{\cot^2 x} + 1 ?
$$
Then you can finish off the whole thing in one more line of simple algebra (plus the identity ##\cos^2 x + \sin^2 x = 1##).
 
  • #4
Ray Vickson said:
Why don't you use the fact that
$$
\frac{1 +\cot^2 x}{\cot^2 x} = \frac{1}{\cot^2 x} + 1 ?
$$
Then you can finish off the whole thing in one more line of simple algebra (plus the identity ##\cos^2 x + \sin^2 x = 1##).

so if 1 / cot^2x = tan^2x = sin^2x / cos^2x

Then all we have to do is: = (sin^2x/cos^2x ) + (cos^2x/cos^2x)
=( sin^2x + cos^2x)/ cos^2x
= 1 / cos^2x

Thanks so much haha, I didn't even notice that. Also another question for proofs. We know that sin^2x + cos^2x = 1 (pythagorean identity). Can we say the same for sinx + cosx= 1? I am pretty sure not but just double checking.
 
  • #5
Jen23 said:
so if 1 / cot^2x = tan^2x = sin^2x / cos^2x

Then all we have to do is: = (sin^2x/cos^2x ) + (cos^2x/cos^2x)
=( sin^2x + cos^2x)/ cos^2x
= 1 / cos^2x

Thanks so much haha, I didn't even notice that. Also another question for proofs. We know that sin^2x + cos^2x = 1 (pythagorean identity). Can we say the same for sinx + cosx= 1? I am pretty sure not but just double checking.

Absolutely not. Try it if ##x=\pi##.
 
  • #6
The brackets [ ] on your second line are incorrect
 

Related to How Do You Simplify Trigonometric Expressions Using Basic Identities?

1. What are trigonometric identities?

Trigonometric identities are mathematical equations that involve trigonometric functions and are true for all values of the variables in the equation. They are used to simplify expressions and solve problems involving triangles and angles.

2. Why are trigonometric identities important?

Trigonometric identities are important because they allow us to simplify complex expressions and solve problems involving angles and triangles. They are also used in various fields such as physics, engineering, and navigation.

3. How many trigonometric identities are there?

There are several fundamental trigonometric identities, but there are an infinite number of trigonometric identities that can be derived from them. Some common identities include the Pythagorean identities, double angle identities, and sum and difference identities.

4. How do you prove trigonometric identities?

There are several methods for proving trigonometric identities, including using algebraic manipulation, using geometric proofs, and using the unit circle. The key is to use the known identities and properties of trigonometric functions to transform one side of the equation into the other.

5. What are some real-world applications of trigonometric identities?

Trigonometric identities have many real-world applications, such as in navigation to determine distances and angles, in physics to analyze the motion of objects, and in engineering to design and construct structures. They are also used in astronomy, architecture, and surveying.

Similar threads

  • Precalculus Mathematics Homework Help
Replies
6
Views
2K
  • Precalculus Mathematics Homework Help
Replies
1
Views
1K
  • Precalculus Mathematics Homework Help
Replies
6
Views
1K
  • Precalculus Mathematics Homework Help
Replies
18
Views
2K
  • Precalculus Mathematics Homework Help
Replies
7
Views
1K
  • Precalculus Mathematics Homework Help
Replies
2
Views
2K
  • Precalculus Mathematics Homework Help
Replies
7
Views
2K
  • Precalculus Mathematics Homework Help
Replies
4
Views
618
  • Precalculus Mathematics Homework Help
Replies
8
Views
1K
  • Precalculus Mathematics Homework Help
Replies
10
Views
2K
Back
Top