What is the solution to the equation tan²x + cos2x = 1 - cos2xtan²x?

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Homework Help Overview

The problem involves the equation tan²x + cos2x = 1 - cos2xtan²x, which is situated within the context of trigonometric identities and equations.

Discussion Character

  • Exploratory, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss transforming the equation using trigonometric identities, such as expressing cos(2x) in terms of sine and cosine. There are attempts to manipulate the equation by clearing denominators and rearranging terms.

Discussion Status

Some participants have made progress in rewriting the equation but express uncertainty about the next steps. Guidance has been offered to multiply out terms and clear fractions, indicating a collaborative effort to explore the problem further.

Contextual Notes

There is mention of difficulties in progressing with the solution, and some participants note corrections to their previous statements, reflecting the iterative nature of the discussion.

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Homework Statement



tan^2x + cos2x =1 - cos2xtan^2x

Homework Equations





The Attempt at a Solution


I have not really gotten anywhere.
 
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Use cos(2x)=cos(x)^2-sin(x)^2 and tan(x)=sin(x)/cos(x) to turn everything into sines and cosines. Then clear out the denominators and start rearranging things.
 
I have gotten it to (sin(x)^2/cos(x)^2) + (cos(x)^2- sin(x)^2) = 1- (cos (x)^2 - sin(x)^2)(sin(x)^2/cos(x)^2)

not sure how to proceed
 
Good so far. Multiply out the right side and multiply both sides by cos(x)^2 to clear out the fractions. Does anything cancel? Rearrange what's left.
 
Its just not happenin for me right now

I have it at sin(x)^2 cos(x)^2(cos(x)^2 - sin(x)^2) = cos(x)^2 - (cos(x)^2- sin(x)^2) sin(x)^2
 
I meant to put a + after the first sin(x)^2
 
Just keep going. Multiply out the terms with parentheses.
 

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