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Trig identities again. I don’t get this?? The one with 2 stars is apparently harder but I don’t even get the easy one. 

Guest Oct 1, 2018
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This doesn't involve proving trig identities, but using them to find the value(s) of \(\theta\)  and x that make the equations true.

 

By definition: \(\sec\theta=\frac{1}{\cos\theta}\)

 

Using this in a and rearranging:   \(\frac{1}{\cos^2\theta}=2\tan\theta\)

 

Multiply both sides by \(\cos^2\theta\) and using the fact that \(\tan\theta=\frac{\sin\theta}{\cos\theta}\) we get, on simplifying: \(1=2\sin\theta\cos\theta\)

 

The RHS is just \(\sin{2\theta}\)  so we have \(1=\sin{2\theta}\)

 

This results in \(2\theta=\frac{\pi}{2}\text{ or }\theta=\frac{\pi}{4}\) 

 

 

For b rewrite x as \(2\theta\) and use \(\cos{2\theta}=\cos^2\theta-\sin^2\theta\text{ and }\sin{}2\theta=2\sin\theta\cos\theta\text{ together with }\cos^2\theta+\sin^2\theta=1\)

 

You should eventually find there are two values of x between 0 and 2pi that satisfy the equation.

Alan  Oct 2, 2018
edited by Alan  Oct 2, 2018

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