4.20 Derivatives and Differential Equations4.22 Infinite Products and Partial Fractions

§4.21 Identities

Contents

§4.21(i) Addition Formulas

4.21.1 \mathop{\sin\/}\nolimits u\pm\mathop{\cos\/}\nolimits u=\sqrt{2}\mathop{\sin\/}\nolimits\!\left(u\pm\tfrac{1}{4}\pi\right)=\sqrt{2}\mathop{\cos\/}\nolimits\!\left(u\mp\tfrac{1}{4}\pi\right).
4.21.2 \mathop{\sin\/}\nolimits\!\left(u\pm v\right)=\mathop{\sin\/}\nolimits u\mathop{\cos\/}\nolimits v\pm\mathop{\cos\/}\nolimits u\mathop{\sin\/}\nolimits v,
4.21.3 \mathop{\cos\/}\nolimits\!\left(u\pm v\right)=\mathop{\cos\/}\nolimits u\mathop{\cos\/}\nolimits v\mp\mathop{\sin\/}\nolimits u\mathop{\sin\/}\nolimits v,
4.21.4 \mathop{\tan\/}\nolimits\!\left(u\pm v\right)=\frac{\mathop{\tan\/}\nolimits u\pm\mathop{\tan\/}\nolimits v}{1\mp\mathop{\tan\/}\nolimits u\mathop{\tan\/}\nolimits v},
4.21.5 \mathop{\cot\/}\nolimits\!\left(u\pm v\right)=\frac{\pm\mathop{\cot\/}\nolimits u\mathop{\cot\/}\nolimits v-1}{\mathop{\cot\/}\nolimits u\pm\mathop{\cot\/}\nolimits v}.
4.21.6 \mathop{\sin\/}\nolimits u+\mathop{\sin\/}\nolimits v=2\mathop{\sin\/}\nolimits\!\left(\frac{u+v}{2}\right)\mathop{\cos\/}\nolimits\!\left(\frac{u-v}{2}\right),
4.21.7 \mathop{\sin\/}\nolimits u-\mathop{\sin\/}\nolimits v=2\mathop{\cos\/}\nolimits\!\left(\frac{u+v}{2}\right)\mathop{\sin\/}\nolimits\!\left(\frac{u-v}{2}\right),
4.21.8 \mathop{\cos\/}\nolimits u+\mathop{\cos\/}\nolimits v=2\mathop{\cos\/}\nolimits\!\left(\frac{u+v}{2}\right)\mathop{\cos\/}\nolimits\!\left(\frac{u-v}{2}\right),
4.21.9 \mathop{\cos\/}\nolimits u-\mathop{\cos\/}\nolimits v=-2\mathop{\sin\/}\nolimits\!\left(\frac{u+v}{2}\right)\mathop{\sin\/}\nolimits\!\left(\frac{u-v}{2}\right).
4.21.10 \mathop{\tan\/}\nolimits u\pm\mathop{\tan\/}\nolimits v=\frac{\mathop{\sin\/}\nolimits\!\left(u\pm v\right)}{\mathop{\cos\/}\nolimits u\mathop{\cos\/}\nolimits v},
4.21.11 \mathop{\cot\/}\nolimits u\pm\mathop{\cot\/}\nolimits v=\frac{\mathop{\sin\/}\nolimits\!\left(v\pm u\right)}{\mathop{\sin\/}\nolimits u\mathop{\sin\/}\nolimits v}.

§4.21(ii) Squares and Products

4.21.12 {\mathop{\sin\/}\nolimits^{{2}}}z+{\mathop{\cos\/}\nolimits^{{2}}}z=1,
4.21.13 {\mathop{\sec\/}\nolimits^{{2}}}z=1+{\mathop{\tan\/}\nolimits^{{2}}}z,
4.21.14 {\mathop{\csc\/}\nolimits^{{2}}}z=1+{\mathop{\cot\/}\nolimits^{{2}}}z.
4.21.15 2\mathop{\sin\/}\nolimits u\mathop{\sin\/}\nolimits v=\mathop{\cos\/}\nolimits\!\left(u-v\right)-\mathop{\cos\/}\nolimits\!\left(u+v\right),
4.21.16 2\mathop{\cos\/}\nolimits u\mathop{\cos\/}\nolimits v=\mathop{\cos\/}\nolimits\!\left(u-v\right)+\mathop{\cos\/}\nolimits\!\left(u+v\right),
4.21.17 2\mathop{\sin\/}\nolimits u\mathop{\cos\/}\nolimits v=\mathop{\sin\/}\nolimits\!\left(u-v\right)+\mathop{\sin\/}\nolimits\!\left(u+v\right).
4.21.18 {\mathop{\sin\/}\nolimits^{{2}}}u-{\mathop{\sin\/}\nolimits^{{2}}}v=\mathop{\sin\/}\nolimits\!\left(u+v\right)\mathop{\sin\/}\nolimits\!\left(u-v\right),
4.21.19 {\mathop{\cos\/}\nolimits^{{2}}}u-{\mathop{\cos\/}\nolimits^{{2}}}v=-\mathop{\sin\/}\nolimits\!\left(u+v\right)\mathop{\sin\/}\nolimits\!\left(u-v\right),
4.21.20 {\mathop{\cos\/}\nolimits^{{2}}}u-{\mathop{\sin\/}\nolimits^{{2}}}v=\mathop{\cos\/}\nolimits\!\left(u+v\right)\mathop{\cos\/}\nolimits\!\left(u-v\right).

§4.21(iii) Multiples of the Argument

4.21.21 \mathop{\sin\/}\nolimits\frac{z}{2}=\pm\left(\frac{1-\mathop{\cos\/}\nolimits z}{2}\right)^{{1/2}},
4.21.22 \mathop{\cos\/}\nolimits\frac{z}{2}=\pm\left(\frac{1+\mathop{\cos\/}\nolimits z}{2}\right)^{{1/2}},
4.21.23 \mathop{\tan\/}\nolimits\frac{z}{2}=\pm\left(\frac{1-\mathop{\cos\/}\nolimits z}{1+\mathop{\cos\/}\nolimits z}\right)^{{1/2}}=\frac{1-\mathop{\cos\/}\nolimits z}{\mathop{\sin\/}\nolimits z}=\frac{\mathop{\sin\/}\nolimits z}{1+\mathop{\cos\/}\nolimits z}.

In (4.21.21)–(4.21.23) Table 4.16.1 and analytic continuation will assist in resolving sign ambiguities.

4.21.24 \mathop{\sin\/}\nolimits\!\left(-z\right)=-\mathop{\sin\/}\nolimits z,
4.21.25 \mathop{\cos\/}\nolimits\!\left(-z\right)=\mathop{\cos\/}\nolimits z,
4.21.26 \mathop{\tan\/}\nolimits\!\left(-z\right)=-\mathop{\tan\/}\nolimits z.
4.21.27 \mathop{\sin\/}\nolimits\!\left(2z\right)=2\mathop{\sin\/}\nolimits z\mathop{\cos\/}\nolimits z=\frac{2\mathop{\tan\/}\nolimits z}{1+{\mathop{\tan\/}\nolimits^{{2}}}z},
4.21.28 \mathop{\cos\/}\nolimits\!\left(2z\right)=2{\mathop{\cos\/}\nolimits^{{2}}}z-1=1-2{\mathop{\sin\/}\nolimits^{{2}}}z={\mathop{\cos\/}\nolimits^{{2}}}z-{\mathop{\sin\/}\nolimits^{{2}}}z=\frac{1-{\mathop{\tan\/}\nolimits^{{2}}}z}{1+{\mathop{\tan\/}\nolimits^{{2}}}z},
4.21.29 \mathop{\tan\/}\nolimits\!\left(2z\right)=\frac{2\mathop{\tan\/}\nolimits z}{1-{\mathop{\tan\/}\nolimits^{{2}}}z}=\frac{2\mathop{\cot\/}\nolimits z}{{\mathop{\cot\/}\nolimits^{{2}}}z-1}=\frac{2}{\mathop{\cot\/}\nolimits z-\mathop{\tan\/}\nolimits z}.
4.21.30 \mathop{\sin\/}\nolimits\!\left(3z\right)=3\mathop{\sin\/}\nolimits z-4{\mathop{\sin\/}\nolimits^{{3}}}z,
4.21.31 \mathop{\cos\/}\nolimits\!\left(3z\right)=-3\mathop{\cos\/}\nolimits z+4{\mathop{\cos\/}\nolimits^{{3}}}z,
4.21.32 \mathop{\sin\/}\nolimits\!\left(4z\right)=8{\mathop{\cos\/}\nolimits^{{3}}}z\mathop{\sin\/}\nolimits z-4\mathop{\cos\/}\nolimits z\mathop{\sin\/}\nolimits z,
4.21.33 \mathop{\cos\/}\nolimits\!\left(4z\right)=8{\mathop{\cos\/}\nolimits^{{4}}}z-8{\mathop{\cos\/}\nolimits^{{2}}}z+1.

De Moivre’s Theorem

When n\in\Integer

4.21.34 \mathop{\cos\/}\nolimits\!\left(nz\right)+i\mathop{\sin\/}\nolimits\!\left(nz\right)=(\mathop{\cos\/}\nolimits z+i\mathop{\sin\/}\nolimits z)^{n}.

This result is also valid when n is fractional or complex, provided that -\pi\leq\realpart{z}\leq\pi.

4.21.35 \mathop{\sin\/}\nolimits\!\left(nz\right)=2^{{n-1}}\prod _{{k=0}}^{{n-1}}\mathop{\sin\/}\nolimits\!\left(z+\frac{k\pi}{n}\right), n=1,2,3,\dots.

If t=\mathop{\tan\/}\nolimits\!\left(\frac{1}{2}z\right), then

4.21.36
\mathop{\sin\/}\nolimits z=\frac{2t}{1+t^{2}},
\mathop{\cos\/}\nolimits z=\frac{1-t^{2}}{1+t^{2}},
dz=\frac{2}{1+t^{2}}dt.

§4.21(iv) Real and Imaginary Parts; Moduli

With z=x+iy

4.21.37 \mathop{\sin\/}\nolimits z=\mathop{\sin\/}\nolimits x\mathop{\cosh\/}\nolimits y+i\mathop{\cos\/}\nolimits x\mathop{\sinh\/}\nolimits y,
4.21.38 \mathop{\cos\/}\nolimits z=\mathop{\cos\/}\nolimits x\mathop{\cosh\/}\nolimits y-i\mathop{\sin\/}\nolimits x\mathop{\sinh\/}\nolimits y,
4.21.39 \mathop{\tan\/}\nolimits z=\frac{\mathop{\sin\/}\nolimits\!\left(2x\right)+i\mathop{\sinh\/}\nolimits\!\left(2y\right)}{\mathop{\cos\/}\nolimits\!\left(2x\right)+\mathop{\cosh\/}\nolimits\!\left(2y\right)},
4.21.40 \mathop{\cot\/}\nolimits z=\frac{\mathop{\sin\/}\nolimits\!\left(2x\right)-i\mathop{\sinh\/}\nolimits\!\left(2y\right)}{\mathop{\cosh\/}\nolimits\!\left(2y\right)-\mathop{\cos\/}\nolimits\!\left(2x\right)}.
4.21.41 |\mathop{\sin\/}\nolimits z|=({\mathop{\sin\/}\nolimits^{{2}}}x+{\mathop{\sinh\/}\nolimits^{{2}}}y)^{{1/2}}=\left(\tfrac{1}{2}\left(\mathop{\cosh\/}\nolimits\!\left(2y\right)-\mathop{\cos\/}\nolimits\!\left(2x\right)\right)\right)^{{1/2}},
4.21.42 |\mathop{\cos\/}\nolimits z|=({\mathop{\cos\/}\nolimits^{{2}}}x+{\mathop{\sinh\/}\nolimits^{{2}}}y)^{{1/2}}=\left(\tfrac{1}{2}(\mathop{\cosh\/}\nolimits\!\left(2y\right)+\mathop{\cos\/}\nolimits\!\left(2x\right))\right)^{{1/2}},
4.21.43 |\mathop{\tan\/}\nolimits z|=\left(\frac{\mathop{\cosh\/}\nolimits\!\left(2y\right)-\mathop{\cos\/}\nolimits\!\left(2x\right)}{\mathop{\cosh\/}\nolimits\!\left(2y\right)+\mathop{\cos\/}\nolimits\!\left(2x\right)}\right)^{{1/2}}.