I'm stuck on this.I need to simplify this given expression $$\cot(\fracπ4+\theta)(\cos^2\theta-\sin^2\theta)$$ So far I simplified the first parentheses to $\frac{\cos\theta-\sin\theta}{\cos\theta\sin\theta}$ which I dont if this is correct but now I dont know how to multiply this by the second parenthesis
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1$\begingroup$ Let us continue what you started, writing $c$ for $\cos\theta$ and $s$ for $\sin\theta$: you already know the expression is $$A=\frac{c-s}{c+s}(c^2-s^2)$$ hence $$A=\frac{c-s}{c+s}(c-s)(c+s)=(c-s)^2=c^2+s^2-2cs.$$ Now, $c^2+s^2=1$ and $2cs=\sin(2\theta)$ hence $$A=1-2\sin\theta\cos\theta=1-\sin(2\theta).$$ $\endgroup$– DidCommented Nov 29, 2015 at 17:09
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$\begingroup$ @SimpleArt Yes I am sure I "did that right". Would you be kind enough to explain what is "by far incorrect" here? $\endgroup$– DidCommented Jan 5, 2016 at 0:28
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$\begingroup$ @Did Oh my bad. It wasn't very obvious with your $c's$ and $s's$. $\endgroup$– Simply Beautiful ArtCommented Jan 5, 2016 at 0:47
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$\begingroup$ You ought to put it in as an answer, not as a comment (in my opinion). $\endgroup$– Simply Beautiful ArtCommented Jan 5, 2016 at 0:47
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1 Answer
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Remember this:
$$\cos^2(x) - \sin^2(x) = \cos(2x)$$
and $$\cot\left(\frac{\pi}{4} - x \right) = \tan\left(\frac{\pi}{4} + x\right)$$
answered Nov 29, 2015 at 15:20
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1$\begingroup$ OK, remembered. What is next? In other words: please explain how this is an answer. $\endgroup$– DidCommented Nov 29, 2015 at 17:11