\(\sec\left(-\dfrac\pi{12}\right)\)
First use the fact that \(\sec x\) is an even function, i.e. \(\sec(-x)=\sec x\). So this is the same as
\(\sec\dfrac\pi{12}\)
Now, note that \(\dfrac\pi{12}=\dfrac\pi3-\dfrac\pi4\), so
\(\sec\dfrac\pi{12}=\sec\left(\dfrac\pi3-\dfrac\pi4\right)=\dfrac1{\cos\frac\pi3\cos\frac\pi4+\sin\frac\pi3\sin\frac\pi4}\)
which comes from the sum identity for cosine,
\(\cos(x\pm y)=\cos x\cos y\mp\sin x\sin y\)
Now,
\(\sec\dfrac\pi{12}=\dfrac1{\frac12\frac1{\sqrt2}+\frac{\sqrt3}2\frac1{\sqrt2}}=\sqrt6-\sqrt2\)
