13. Other sums of trigonometric functions

Sum of sines and cosines with arguments in arithmetic progression:^] if α ≠ 0, then

${\displaystyle {\begin{aligned}&\sin \varphi +\sin(\varphi +\alpha )+\sin(\varphi +2\alpha )+\cdots \\[8pt]&{}\qquad \qquad \cdots +\sin(\varphi +n\alpha )={\frac {\sin {\frac {(n+1)\alpha }{2}}\cdot \sin(\varphi +{\frac {n\alpha }{2}})}{\sin {\frac {\alpha }{2}}}}\quad {\text{and}}\\[10pt]&\cos \varphi +\cos(\varphi +\alpha )+\cos(\varphi +2\alpha )+\cdots \\[8pt]&{}\qquad \qquad \cdots +\cos(\varphi +n\alpha )={\frac {\sin {\frac {(n+1)\alpha }{2}}\cdot \cos(\varphi +{\frac {n\alpha }{2}})}{\sin {\frac {\alpha }{2}}}}.\end{aligned}}}$

For any a and b:

${\displaystyle a\cos x+b\sin x={\sqrt {a^{2}+b^{2}}}\cos {\big (}x-\operatorname {atan2} \,(b,a){\big )}\;}$

where atan2(y, x) is the generalization of arctan(y/x) that covers the entire circular range.

${\displaystyle \sec x\pm \tan x=\tan \left({\frac {\pi }{4}}\pm {\frac {x}{2}}\right).}$

The above identity is sometimes convenient to know when thinking about the Gudermannian function, which relates the circular and hyperbolic trigonometric functions without resorting to complex numbers.

If x, y, and z are the three angles of any triangle, i.e. if x + y + z = π, then

${\displaystyle \cot x\cot y+\cot y\cot z+\cot z\cot x=1.\,}$