Trigonometry

Sine and cosine
$$sin(a)^{2}+cos(a)^{2} = 1$$
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      =   
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Tangent
$$tg(a) = \frac{sin(a)}{cos(a)}$$
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Cotangent
$$ctg(a) = \frac{cos(a)}{sin(a)}$$
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Product of tangent and cotangent
$$tg(a)\cdot ctg(a) = 1$$
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Tangent and cosine
$$1+tg(a)^{2} = \frac{1}{cos(a)^{2}}$$
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Cotangent and sine
$$1+ctg(a)^{2} = \frac{1}{sin(a)^{2}}$$
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Sine of sum of angles
$$sin(a+b) = sin(a)\cdot cos(b)+cos(a)\cdot sin(b)$$
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Sine of difference of angles
$$sin(a-b) = sin(a)\cdot cos(b)-cos(a)\cdot sin(b)$$
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Cosine of sum of angles
$$cos(a+b) = cos(a)\cdot cos(b)-sin(a)\cdot sin(b)$$
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Cosine of difference of angles
$$cos(a-b) = cos(a)\cdot cos(b)+sin(a)\cdot sin(b)$$
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Tangent of sum of angles
$$tg(a+b) = \frac{tg(a)+tg(b)}{1-tg(a)\cdot tg(b)}$$
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Tangent of difference of angles
$$tg(a-b) = \frac{tg(a)-tg(b)}{1+tg(a)\cdot tg(b)}$$
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Sine of double angle
$$sin(2\cdot a) = 2\cdot sin(a)\cdot cos(a)$$
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Cosine of double angle
$$cos(2\cdot a) = cos(a)^{2}-sin(a)^{2}$$
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Cosine of double angle
$$cos(2\cdot a) = 1-2\cdot sin(a)^{2}$$
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Cosine of double angle
$$cos(2\cdot a) = 2\cdot cos(a)^{2}-1$$
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Tangent of double angle
$$tg(2\cdot a) = \frac{2\cdot tg(a)}{1-tg(a)^{2}}$$
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Cotangent of double angle
$$ctg(2\cdot a) = \frac{1-tg(a)^{2}}{2\cdot tg(a)}$$
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Cotangent of double angle
$$ctg(2\cdot a) = \frac{ctg(a)-tg(a)}{2}$$
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Sum of sines (sum to product)
$$sin(a)+sin(b) = 2\cdot sin(\frac{a+b}{2})\cdot cos(\frac{a-b}{2})$$
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Difference of sines (difference to product)
$$sin(a)-sin(b) = 2\cdot cos(\frac{a+b}{2})\cdot sin(\frac{a-b}{2})$$
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Sum of cosines (sum to product)
$$cos(a)+cos(b) = 2\cdot cos(\frac{a+b}{2})\cdot cos(\frac{a-b}{2})$$
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Difference of cosines (difference to product)
$$cos(a)-cos(b) = -2\cdot sin(\frac{a+b}{2})\cdot sin(\frac{a-b}{2})$$
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Product of sine and cosine
$$sin(a)\cdot cos(b) = \frac{1}{2}\cdot (sin(a-b)+sin(a+b))$$
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Product of sines
$$sin(a)\cdot sin(b) = \frac{1}{2}\cdot (cos(a-b)-cos(a+b))$$
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Product of cosines
$$cos(a)\cdot cos(b) = \frac{1}{2}\cdot (cos(a-b)+cos(a+b))$$
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Power reduction for sine
$$sin(a)^{2} = \frac{1-cos(2\cdot a)}{2}$$
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Power reduction for cosine
$$cos(a)^{2} = \frac{1+cos(2\cdot a)}{2}$$
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