Solve the trigonometric expression \[\dfrac{{\sin A - \sin 5A + \sin 9A - \sin 13A}}{{\cos A - \cos 5A - \cos 9A + \cos 13A}} = \cot 4A\]

Hint: We will use the basic trigonometric properties of sine and cosine angles accordingly at both the numerator and denominator after rearranging them, and then we simplify it to simpler terms to get our desired right hand side.

Complete step-by-step solution:
Now from our previous knowledge of trigonometry and its formulas we have,
\[
  \sin A - \sin B = 2\cos (\dfrac{{A + B}}{2})\sin (\dfrac{{A - B}}{2}) \\
  \cos A + \cos B = 2\cos (\dfrac{{A + B}}{2})\cos (\dfrac{{A - B}}{2}) \]
Now let us simplify the terms in our left hand side using the properties above to get,
\[\dfrac{{(\sin A - \sin 13A) - (\sin 5A - \sin 9A)}}{{(\cos A + \cos 13A) - (\cos 5A + \cos 9A)}}\]
\[ = \dfrac{{2\cos (\dfrac{{13A + A}}{2})\sin (\dfrac{{A - 13A}}{2}) - 2\cos (\dfrac{{9A + 5A}}{2})\sin (\dfrac{{5A - 9A}}{2})}}{{2\cos (\dfrac{{A + 13A}}{2})\cos (\dfrac{{A - 13A}}{2}) - 2\cos (\dfrac{{5A + 9A}}{2})\cos (\dfrac{{5A - 9A}}{2})}}\]
\[ = \dfrac{{2\cos (7A)\sin ( - 6A) - 2\cos (7A)\sin ( - 2A)}}{{2\cos (7A)\cos ( - 6A) - 2\cos (7A)\cos ( - 2A)}}\]
We know also that,
 \[ \sin ( - A) = - \sin A \\
  \cos ( - A) = \cos A \]
\[ = \dfrac{{ - 2\cos (7A)\sin (6A) + 2\cos (7A)\sin (2A)}}{{2\cos (7A)\cos (6A) - 2\cos (7A)\cos (2A)}} \\
   = \dfrac{{2\cos (7A)(\sin (2A) - \sin (6A))}}{{2\cos (7A)(\cos (6A) - \cos (2A))}} \\
   = \dfrac{{\sin (2A) - \sin (6A)}}{{\cos (6A) - \cos (2A)}} \\
   = \dfrac{{2\cos (\dfrac{{6A + 2A}}{2})\sin (\dfrac{{2A - 6A}}{2})}}{{ - 2\sin (\dfrac{{6A + 2A}}{2})\sin (\dfrac{{6A - 2A}}{2})}} \\
   = \dfrac{{ - \cos 4A.\sin 2A}}{{ - \sin 4A\sin 2A}} \\
   = \cot 4A \]
Therefore, we get the desired result i.e. the right hand side.
Additional information: In mathematics, the trigonometric functions also called circular functions, angle functions or goniometric functions are real functions which relate to the angle of a right-angled triangle in terms of two side lengths. They are widely used in all sciences, and with respect to geometry, celestial mechanics, and many, many more. The most widely used trigonometric functions are the modern mathematics, the sine, cosine, and tangent. The links between them are respectively the cosecant, secant. Each of the six trigonometric functions has a corresponding inverse function that is the inverse of the trigonometric function. The tangent function is one of the most important trigonometric functions. A tangent line is defined as the ratio of the opposite side to the adjacent side of a given angle in a right-angled triangle. Cosine or cos function is a periodic function with a period of \[2\pi \]. The domain of cosine function is from \[( - \infty ,\infty )\] and the range is \[[ - 1,1]\]. The graph of the cosine function is symmetric about the y-axis and it is an even function. Also the minimum value of \[\cos \theta \] occurs when \[\theta = \pi + 2n\pi \] where \[n\] is an integer.

Note: It is important that we know the basic properties of sine and cosine function and their relationships with tan and cot functions. It is also important where the functions are positive and negative and then approach accordingly.