0
$\begingroup$

There are two terms I have trouble with simplifying:

1.)

$\dfrac{\tan^2(\gamma) -\cot^2(\gamma)}{4(2\sin^2(\gamma) - 1)} = ?$

The result is supposed to be $\dfrac{1}{4}$.

2.)

$2\sin^2(\delta) + \cos^4(\delta) - \sin^4(\delta) = ?$

For this one the result is supposed to be $1$.

How do I simplify these terms?

  • 0
    Oh well, I'm glad it turned out to be wrong, so that I don't have to beat myself for not getting the "expected" result...2012-05-20

1 Answers 1

2

For the first one, I don't think it can be simplified more than to $\frac{1}{4}\sec\gamma\csc\gamma=\csc^2(2\gamma)$, and it certainly won't simplify to $\frac{1}{4}$ (try substituting any value for $\gamma$, and you'll see that the result varies with $\gamma$, hence it can't simplify to a constant).

For the second one write it like TMM suggested and use the pythagorean identity $\cos^2{\delta}+\sin^2\delta=1$:

$ \begin{align} 2\sin^2\delta + \cos^4\delta - \sin^4\delta =& 2\sin^2\delta+(\cos^2\delta-\sin^2\delta)(\cos^2\delta+\sin^2\delta)\\ =&\cos^2\delta+\sin^2\delta\\ =&1 \end{align} $

  • 0
    @MiroslavCetojevic Intentional. Note that $\sec\gamma=\frac{1}{\cos\gamma}$ and $\csc\gamma=\frac{1}{\sin\gamma}$.2012-05-20