Derivatives of Trigonometric functions with Multiple angles

In calculus, six differentiation formulas are developed to find the derivatives of the trigonometric functions but they are not useful in the case of finding the derivatives of the trigonometric functions, which contain either multiple or submultiple angles. Hence, we should develop the differentiation laws for the trigonometric functions in which multiple or sub multiple angles are involved.

The following six derivative rules are the multiple-angle trigonometric differentiation properties with proofs to find the derivatives of the trigonometric functions, which consist of multiple or sub-multiple angles.

1. $\dfrac{d}{dx}{\,\sin{ax}}$ $\,=\,$ $a\cos{ax}$Sine$1$ is a factor of itself.
2. $\dfrac{d}{dx}{\,\cos{ax}} \,=\, -a\sin{ax}$Cosine$1$ and $2$ are factors of $2$.
3. $\dfrac{d}{dx}{\,\tan{ax}} \,=\, a\sec^2{ax}$Tangent$1$ and $3$ are factors of $3$.
4. $\dfrac{d}{dx}{\,\cot{ax}}$ $\,=\,$ $-a\csc^2{ax}$Cotangent$1,$ $2$ and $4$ are factors of $4$.
5. $\dfrac{d}{dx}{\,\sec{ax}} \,=\, a\sec{ax}\tan{ax}$Secant$1$ and $5$ are factors of $5$.
6. $\dfrac{d}{dx}{\,\csc{ax}}$ $\,=\,$ $-a\csc{ax}\cot{ax}$Cosecant$1,$ $2,$ $3$ and $6$ are factors of $6$.