Home page for accesible maths 1 1 Further Integration

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1.39 Useful identities

cosh2⁡u-1=sinh2⁡u,  sec2⁡u=1+tan2⁡u,\cosh^{2}u-1=\sinh^{2}u,\qquad\sec^{2}u=1+\tan^{2}u,
cos⁡u=±11+tan2⁡u,  sin⁡u=±tan⁡u1+tan2⁡u.\cos u={{\pm 1}\over{\sqrt{1+\tan^{2}u}}},\qquad\sin u={{\pm\tan u}\over{\sqrt% {1+\tan^{2}u}}}.

Consider the right-angled triangle with angle uu that has adjacent 11, opposite xx and hypotenuse 1+x2\sqrt{1+x^{2}}.

Then tan⁡u=x\tan u=x, sin⁡u=x/1+x2\sin u=x/\sqrt{1+x^{2}} , cos⁡u=1/1+x2\cos u=1/\sqrt{1+x^{2}}.

Trigonometric or hyperbolic? The choice partly depends upon the user’s preference; but some substitutions can make an integral more difficult. If a substitution does not help, then try another one.