<s>[INST] Convert $f(x) = (\cos x+\sin x)(1-\sqrt{3}\tan x)$ into the form of $k(1+\tan x)\cos(x+a).$ I need your help to convert the following into the form of $k(1+\tan(x))\cos(x+a).$
$$f(x) = (\cos(x)+\sin(x))(1-\sqrt{3}\tan x)$$
Here is what I've done so far.
\begin{aligned}
f(x) &= (\cos(x)+\sin(x))(1-\sqrt{3}\tan x)\\
f(x) &= \sqrt{2}(\frac{1}{\sqrt{2}}\cos(x)+\frac{1}{\sqrt{2}}\sin(x))(1-\sqrt{3}\tan x)\\
f(x) &= \sqrt{2}\cos(x-\frac{\pi}{4})(1-\sqrt{3}\tan x)
\end{aligned}
Please help me.
 [/INST] We have to find $k$ and $a$ such that
$k(1+\tan x)\cos(x+a)=(\cos x+\sin x)(1-\sqrt3\tan x)$
or
$k(\cos x+\sin x)\dfrac{\cos(x+a)}{\cos x}=(\cos x+\sin x)(1-\sqrt3\tan x)$
or
$k(\cos a-\sin a\tan x)=(1-\sqrt3\tan x).$
Now the similarity is obvious. Can you take it from here?
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