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.

The derivative of sine of a multiple angle with respect to a variable is equal to the constant times cosine of multiple-angle.

$\dfrac{d}{dx}{\,\sin{(ax)}}$ $\,=\,$ $a\cos{(ax)}$

The derivative of cosine of a multiple angle with respect to a variable is equal to the minus constant times sine of multiple-angle.

$\dfrac{d}{dx}{\,\cos{(ax)}} \,=\, -a\sin{(ax)}$

The derivative of tangent of a multiple angle with respect to a variable is equal to the constant times square of secant of multiple-angle.

$\dfrac{d}{dx}{\,\tan{(ax)}} \,=\, a\sec^2{(ax)}$

The derivative of cotangent of a multiple angle with respect to a variable is equal to the minus constant times square of cosecant of multiple-angle.

$(1).\,\,$ $\dfrac{d}{dx}{\,\cot{(ax)}}$ $\,=\,$ $-a\csc^2{(ax)}$

$(2).\,\,$ $\dfrac{d}{dx}{\,\cot{(ax)}}$ $\,=\,$ $-a\operatorname{cosec}^2{(ax)}$

The derivative of secant of a multiple angle with respect to a variable is equal to the constant times the product of secant and tangent of multiple-angle.

$\dfrac{d}{dx}{\,\sec{(ax)}} \,=\, a\sec{(ax)}.\tan{(ax)}$

The derivative of cosecant of a multiple angle with respect to a variable is equal to the minus constant times the product of cosecant and cotangent of multiple-angle.

$(1).\,\,$ $\dfrac{d}{dx}{\,\csc{(ax)}}$ $\,=\,$ $-a\csc{(ax)}.\cot{(ax)}$

$(2).\,\,$ $\dfrac{d}{dx}{\,\operatorname{cosec}{(ax)}}$ $\,=\,$ $-a\operatorname{cosec}{(ax)}.\cot{(ax)}$

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