MHB How to find a general solution to sec2 2x = 1– tan 2x

AI Thread Summary
To find a general solution for the equation sec²(2x) = 1 - tan(2x), the Pythagorean identity sec²(θ) = tan²(θ) + 1 is applied, leading to the rearranged equation tan²(2x) + tan(2x) = 0. This can be factored into tan(2x)(tan(2x) + 1) = 0, indicating two potential solutions: tan(2x) = 0 or tan(2x) = -1. The discussion highlights the importance of recognizing trigonometric identities in solving such equations. Overall, the collaborative effort successfully guides the user toward the solution.
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Hi all,

My name is Arijit Biswas. I have resumed learning maths after a long time and I am stuck with a simple problem in trigonometry.

I need to find a general solution to the equation: sec2 2x = 1– tan 2x. I have worked out something i.e.
1) Multiply by cos2 2x and that makes the equation to: 1 = cos2 2x - sin 2x.cos 2x
2) 1 = cos 2x (cos 2x - sin 2x)
3) Thereafter I expand all terms but I do not find the solutions

Could anybody please help?

Thanks a lot in advance!

Regards,
Arijit
 
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Hello Arijit!

We are given to solve:

$$\sec^2(2x)=1-\tan(2x)$$

I would begin by applying the Pythagorean identity $$\sec^2(\theta)=\tan^2(\theta)+1$$ so that we now have:

$$\tan^2(2x)+1=1-\tan(2x)$$

Now, arrange as:

$$\tan^2(2x)+\tan(2x)=0$$

Factor:

$$\tan(2x)\left(\tan(2x)+1\right)=0$$

Can you proceed?
 
MarkFL said:
Hello Arijit!

We are given to solve:

$$\sec^2(2x)=1-\tan(2x)$$

I would begin by applying the Pythagorean identity $$\sec^2(\theta)=\tan^2(\theta)+1$$ so that we now have:

$$\tan^2(2x)+1=1-\tan(2x)$$

Now, arrange as:

$$\tan^2(2x)+\tan(2x)=0$$

Factor:

$$\tan(2x)\left(\tan(2x)+1\right)=0$$

Can you proceed?

Hi there,

Thank you so much for the solution. You wouldn't believe, but I was just looking at the Pythagorean identity you used to solve the equation. However it didn't occur to me. How silly! Thanks a lot again!
 
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