Trigonometry/Identities

Let us take a right angled triangle with hypotenuse length 1. If we mark one of the acute angles as $\theta$, then using the definition of the sine ratio, we have

$\sin \theta ={\cfrac {opposite}{hypotenuse}}$

As the hypotenuse is 1,

$\sin \theta ={\cfrac {opposite}{1}}=opposite$

Repeating the same process using the definition of the cosine ratio, we have

$\cos \theta ={\cfrac {adjacent}{hypotenuse}}={\cfrac {adjacent}{1}}=adjacent$

Pythagorean identities

Since this is a right triangle, we can use the Pythagorean Theorem:

$x^{2}+y^{2}=r^{2}$

${\frac {x^{2}}{r^{2}}}+{\frac {y^{2}}{r^{2}}}={\frac {r^{2}}{r^{2}}}$

$\operatorname {cos} ^{2}\theta +\operatorname {sin} ^{2}\theta =1$

This is the most fundamental identity in trigonometry.

${\frac {x^{2}}{y^{2}}}+{\frac {y^{2}}{y^{2}}}={\frac {r^{2}}{y^{2}}}$

$\operatorname {cot} ^{2}x+1=\operatorname {csc} ^{2}$

${\frac {x^{2}}{x^{2}}}+{\frac {y^{2}}{x^{2}}}={\frac {r^{2}}{x^{2}}}$

$\operatorname {1} +\operatorname {tan} ^{2}\theta =\operatorname {sec} ^{2}\theta$

From this identity, if we divide through by squared cosine, we are left with:

${\cfrac {\operatorname {sin} ^{2}\theta +\operatorname {cos} ^{2}\theta }{\operatorname {cos} ^{2}\theta }}={\cfrac {1}{\operatorname {cos} ^{2}\theta }}$

$\operatorname {tan} ^{2}\theta +1=\operatorname {sec} ^{2}\theta$

$\operatorname {sec} ^{2}\theta -\operatorname {tan} ^{2}\theta =1$

If instead we divide the original identity by squared sine, we are left with:

${\cfrac {\operatorname {sin} ^{2}\theta +\operatorname {cos} ^{2}\theta }{\operatorname {sin} ^{2}\theta }}={\cfrac {1}{\operatorname {sin} ^{2}\theta }}$

$\operatorname {cot} ^{2}\theta +1=\operatorname {csc} ^{2}\theta$

$\operatorname {csc} ^{2}\theta -\operatorname {cot} ^{2}\theta =1$

There are basically 3 main trigonometric identities. The proofs come directly from the definitions of these functions and the application of the Pythagorean theorem:

$\operatorname {sin} ^{2}\theta +\operatorname {cos} ^{2}\theta =1$

$\operatorname {sec} ^{2}\theta -\operatorname {tan} ^{2}\theta =1$

$\operatorname {csc} ^{2}\theta -\operatorname {cot} ^{2}\theta =1$

Angle sum-difference identities

$\sin(\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta$

$\cos(\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta$

Cofunction identities

$\cos(90-\theta )=\sin \theta$

$\sec(90-\theta )=\csc \theta$

$\tan(90-\theta )=\cot \theta$

$\sin(90-\theta )=\cos \theta$

$\csc(90-\theta )=\sec \theta$

$\cot(90-\theta )=\tan \theta$

Multiple angle identities

$\cos 2A=\cos ^{2}A-\sin ^{2}A$

$\sin 2A=2\sin A\cos A$

$\sin 2\theta ={\frac {tan2\theta *tan\theta }{tan2\theta -tan\theta }}$

$\cos 2\theta ={\frac {tan\theta }{tan2\theta -tan\theta }}$

$\tan 2\theta =tan\theta ({\frac {1}{cos2\theta }}+1)$

$\tan 2\theta ={\frac {2sin^{2}\theta }{sin2\theta -tan\theta }}$