<s>[INST] Sum and difference formulas for $\csc$ and $\sec$ We all are familiar with the sum and difference formulas for $\sin$ and $\cos$, but is there an analogue for the sum and difference formulas for secant and cosecant?  That is, 
$$\csc (A\pm B) = ?$$ and $$\sec (A \pm B) = ?$$
I tried a variation of the sum and difference formulas, but they were incorrect.  Can it be derived geometrically?
 [/INST] Note 
$\sin^2 A \cos^2 B -\cos^2 A \sin^2 B\\
= \sin^2 A (1-\sin^2 B )-\cos^2 A \sin^2 B\\
= \sin^2 A - (\cos^2 A + \sin^2 A)\sin^2 B 
= \sin^2 A - \sin^2 B \\
 =\frac12(\cos2B-\cos2 A )\\
$
Thus, 
$\csc (A \pm B)
=\frac {1}{\sin A \cos B \pm \cos A \sin B} 
=\frac {\sin A \cos B \mp\cos A \sin B}{\sin^2 A \cos^2 B - \cos^2 A \sin^2 B} 
= \frac {2\sin(A\mp B)}{\cos2B- \sin2 A } 
$
Similarly
$\sec (A \pm B)= \frac {2\cos(A\mp B)}{\cos2B+ \sin2 A } 
$
</s>