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Basic Trig Identities

Reciprocal:

$sin\theta =\frac{1}{csc\theta},\ csc\theta =\frac{1}{sin\theta}$

$cos\theta =\frac{1}{sec\theta},\ sec\theta =\frac{1}{cos\theta}$

$tan\theta =\frac{1}{cot\theta},\ cot\theta =\frac{1}{tan\theta}$

Quotient:

$tan\theta=\frac{sin\theta}{cos\theta}$

$cot\theta=\frac{cos\theta}{sin\theta}$

Pythagorean:

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

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

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

Example:

$Prove\ \frac{tan^{2}x}{1+tan^{2}x}=sin^{2}x$

$\frac{tan^{2}x}{sec^{2}x}=sin^{2}x$

$\frac{\frac{sin^{2}x}{cos^{2}x}}{sec^{2}x}=sin^{2}x$

$\frac{\frac{sin^{2}x}{cos^{2}x}}{\frac{1}{cos^{2}x}}=sin^{2}x$

$\frac{sin^{2}x}{cos^{2}x}\cdot \frac{cos^{2}x}{1}=sin^{2}x$

$sin^{2}x=sin^{2}x\ \checkmark$