삼각함수 곱-합 sinAsinB=12[cos(A−B)−cos(A+B)]\sin A\sin B = \frac{1}{2} \left[ \cos(A-B)-\cos(A+B) \right]sinAsinB=21[cos(A−B)−cos(A+B)] cosAcosB=12[cos(A−B)+cos(A+B)]\cos A\cos B = \frac{1}{2} \left[ \cos(A-B)+\cos(A+B) \right]cosAcosB=21[cos(A−B)+cos(A+B)] sinAcosB=12[sin(A+B)+sin(A−B)]\sin A\cos B = \frac{1}{2} \left[ \sin(A+B)+\sin(A-B) \right]sinAcosB=21[sin(A+B)+sin(A−B)] cosAsinB=12[sin(A+B)−sin(A−B)]\cos A\sin B = \frac{1}{2} \left[ \sin(A+B)-\sin(A-B) \right]cosAsinB=21[sin(A+B)−sin(A−B)] - 역삼각함수 미분 공식 ddx(sin−1x)=11−x2,∣x∣<1\frac{d}{dx}\left(\sin^{-1}x\right) = \frac{1}{\sqrt{1-x^2}}, \qquad |x|<1dxd(sin−1x)=1−x21,∣x∣<1 ddx(cos−1x)=−11−x2,∣x∣<1\frac{d}{dx}\left(\cos^{-1}x\right) = -\frac{1}{\sqrt{1-x^2}}, \qquad |x|<1dxd(cos−1x)=−1−x21,∣x∣<1 ddx(tan−1x)=11+x2,x∈R\frac{d}{dx}\left(\tan^{-1}x\right) = \frac{1}{1+x^2}, \qquad x\in\mathbb{R}dxd(tan−1x)=1+x21,x∈R ddx(cot−1x)=−11+x2,x∈R\frac{d}{dx}\left(\cot^{-1}x\right) = -\frac{1}{1+x^2}, \qquad x\in\mathbb{R}dxd(cot−1x)=−1+x21,x∈R ddx(sec−1x)=1∣x∣x2−1,∣x∣>1\frac{d}{dx}\left(\sec^{-1}x\right) = \frac{1}{|x|\sqrt{x^2-1}}, \qquad |x|>1dxd(sec−1x)=∣x∣x2−11,∣x∣>1 ddx(csc−1x)=−1∣x∣x2−1,∣x∣>1\frac{d}{dx}\left(\csc^{-1}x\right) = -\frac{1}{|x|\sqrt{x^2-1}}, \qquad |x|>1dxd(csc−1x)=−∣x∣x2−11,∣x∣>1 쌍곡함수 (Hyperbolic Functions) sinhx=ex−e−x2\sinh x = \frac{e^x-e^{-x}}{2}sinhx=2ex−e−x coshx=ex+e−x2\cosh x = \frac{e^x+e^{-x}}{2}coshx=2ex+e−x 쌍곡함수 기본 항등식 cosh2x−sinh2x=1\cosh^2 x-\sinh^2 x=1cosh2x−sinh2x=1 1−tanh2x=sech2x1-\tanh^2 x=\operatorname{sech}^2 x1−tanh2x=sech2x 쌍곡함수 미분 공식 ddx(sinhx)=coshx\frac{d}{dx}\left(\sinh x\right) = \cosh xdxd(sinhx)=coshx ddx(coshx)=sinhx\frac{d}{dx}\left(\cosh x\right) = \sinh xdxd(coshx)=sinhx ddx(tanhx)=sech2x\frac{d}{dx}\left(\tanh x\right) = \operatorname{sech}^2 xdxd(tanhx)=sech2x ddx(cothx)=−csch2x\frac{d}{dx}\left(\coth x\right) = -\operatorname{csch}^2 xdxd(cothx)=−csch2x ddx(sechx)=−sechxtanhx\frac{d}{dx}\left(\operatorname{sech}x\right) = -\operatorname{sech}x\tanh xdxd(sechx)=−sechxtanhx ddx(cschx)=−cschxcothx\frac{d}{dx}\left(\operatorname{csch}x\right) = -\operatorname{csch}x\coth xdxd(cschx)=−cschxcothx 역쌍곡함수 미분 공식 ddx(sinh−1x)=11+x2,x∈R\frac{d}{dx}\left(\sinh^{-1}x\right) = \frac{1}{\sqrt{1+x^2}}, \qquad x\in\mathbb{R}dxd(sinh−1x)=1+x21,x∈R ddx(cosh−1x)=1x2−1,x>1\frac{d}{dx}\left(\cosh^{-1}x\right) = \frac{1}{\sqrt{x^2-1}}, \qquad x>1dxd(cosh−1x)=x2−11,x>1 ddx(tanh−1x)=11−x2,∣x∣<1\frac{d}{dx}\left(\tanh^{-1}x\right) = \frac{1}{1-x^2}, \qquad |x|<1dxd(tanh−1x)=1−x21,∣x∣<1 ddx(coth−1x)=11−x2,∣x∣>1\frac{d}{dx}\left(\coth^{-1}x\right) = \frac{1}{1-x^2}, \qquad |x|>1dxd(coth−1x)=1−x21,∣x∣>1 ddx(sech−1x)=−1x1−x2,0<x<1\frac{d}{dx}\left(\operatorname{sech}^{-1}x\right) = -\frac{1}{x\sqrt{1-x^2}}, \qquad 0<x<1dxd(sech−1x)=−x1−x21,0<x<1 ddx(csch−1x)=−1∣x∣1+x2,x≠0\frac{d}{dx}\left(\operatorname{csch}^{-1}x\right) = -\frac{1}{|x|\sqrt{1+x^2}}, \qquad x\neq0dxd(csch−1x)=−∣x∣1+x21,x=0 유도는 직접 쳐해라