# Proving Reciprocal Identities: (secx+1)/(sin2x) = (tanx)/2cosx-2cos2x

• Schaus
In summary, the equations for the homework involve left and right sides, cancelling a cosx from both, and then simplifying. The right hand side is incorrect and needs to be corrected.
Schaus

## Homework Statement

(secx+1)/(sin2x) = (tanx)/2cosx-2cos2x)

## The Attempt at a Solution

Left Side
((1+cosx)/cosx)/2sinxcosx

((1+cosx)/cosx) x (1/2sinxcosx)
cancel the a cosx from both to get
(1/2sinxcosx)
This is all I could manage with left side so I tried right side
Right Side
(sinx/cosx)/2cosx-2cos2x)
I'm stuck here. I've been trying to find something to change the denominator of the Right Side but I can't think of anything that will work. If someone could let me know where I am going wrong it would be greatly appreciated!

Schaus said:
((1+cosx)/cosx) x (1/2sinxcosx)
cancel the a cosx from both to get
(1/2sinxcosx)
What happened to the 1+cos?

Rather than working each side separately, multiply out to get rid of all the denominators.

Schaus
I thought I could cancel a cosx, maybe I cannot. I tried to eliminate the denominator like you said. Here's what I got.
(secx+1)(2cosx-2cos2x) = (Sin2x)(tanx)

((1+cosx)/cosx)(2cosx-2cos2x)=(2sinxcosx)(sinx/cosx)
Expanding
((2cosx-2cos2x+2cos2x-2cos3x)/cosx) = 2
Moving the cosx under left side to the right side and simplifying
2cosx-2cos3x = 2cosx
Does this look right? And if so, where do I go from here?

Schaus said:
((2cosx-2cos2x+2cos2x-2cos3x)/cosx) = 2
Check the right hand side.

Woops. I think I should have gotten 2sin2xcosx
2cosx-2cos3x = 2sin2xcosx
Does this look right?

Schaus said:
Woops. I think I should have gotten 2sin2xcosx
2cosx-2cos3x = 2sin2xcosx
Does this look right?
Yes. Keep simplifying.

2cosx-2cos3x = 2sin2xcosx
2cosx(1-cos2x) = 2sin2xcosx
2cosx(sin2x) = 2sin2xcosx
2cosxsin2x = 2sin2xcosx
Does this work?

Schaus said:
2cosx-2cos3x = 2sin2xcosx
2cosx(1-cos2x) = 2sin2xcosx
2cosx(sin2x) = 2sin2xcosx
2cosxsin2x = 2sin2xcosx
Does this work?
Yes.
Of course, it is not strictly kosher to start with the thing to be proved and deduce a tautology. You need all the steps to be reversible. They are in this case, but it is cleaner to rewrite it in the more persuasive sequence: start with the tautology and deduce the thing to be proved.

Schaus
So I should start with 2sin2xcosx and work backwards? Thank you for all the help by the way.

Schaus said:
Ideally, yes.

Ok, I'll try it. I'm going to have to practice these quite a bit more I think.

## 1. What are reciprocal identities?

Reciprocal identities are trigonometric identities that relate the values of one trigonometric function to the values of its reciprocal function. For example, the reciprocal identity for sine is cosecant, and the reciprocal identity for cosine is secant.

## 2. Why is proving reciprocal identities important?

Proving reciprocal identities is important for solving trigonometric equations and simplifying trigonometric expressions. It also helps in understanding the relationships between different trigonometric functions.

## 3. How do you prove a reciprocal identity?

To prove a reciprocal identity, you must manipulate one side of the equation using algebraic and trigonometric properties until it is equivalent to the other side. You can also use the fundamental reciprocal identities, such as sin²x + cos²x = 1, to prove more complex identities.

## 4. What is the most common method for proving reciprocal identities?

The most common method for proving reciprocal identities is using the fundamental reciprocal identities and algebraic manipulations. This involves breaking down the more complex identity into simpler identities and using substitution and simplification to show that both sides are equivalent.

## 5. What are some tips for proving reciprocal identities?

Some tips for proving reciprocal identities include starting with the side that looks more complicated, using algebraic and trigonometric properties, and being familiar with the fundamental reciprocal identities. It is also helpful to check your work by plugging in values for x to see if both sides of the equation give the same result.

• Precalculus Mathematics Homework Help
Replies
28
Views
4K
• Precalculus Mathematics Homework Help
Replies
3
Views
2K
• Precalculus Mathematics Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Precalculus Mathematics Homework Help
Replies
6
Views
4K
• Precalculus Mathematics Homework Help
Replies
4
Views
6K
• Precalculus Mathematics Homework Help
Replies
11
Views
33K
• Precalculus Mathematics Homework Help
Replies
1
Views
2K
• Precalculus Mathematics Homework Help
Replies
7
Views
2K
• Precalculus Mathematics Homework Help
Replies
2
Views
4K