Trigonometric Ratios and Identities

Basic identities

Pythagorean
$sin^{2} θ + cos^{2} θ = 1,$ $1 + tan^{2} θ = sec^{2} θ,$ $1 + cot^{2} θ = csc^{2} θ$
Allied angles
$sin (π - θ) = sin θ, cos (π - θ) = - cos θ, sin (\frac{π}{2} - θ) = cos θ$
ASTC: all ratios positive in quadrant I, sin in II, tan in III, cos in IV.

Sine and cosine
$sin (A \pm B) = sin A cos B \pm cos A sin B,$ $cos (A \pm B) = cos A cos B \mp sin A sin B$
Tangent
$tan (A \pm B) = \frac{tan A \pm tan B}{1 \mp tan A tan B}$
Products
$sin (A + B) sin (A - B) = sin^{2} A - sin^{2} B,$ $cos (A + B) cos (A - B) = cos^{2} A - sin^{2} B$

Double angle
$sin 2 A = 2 sin A cos A = \frac{2 tan A}{1 + tan ^{2} A},$ $cos 2 A = cos^{2} A - sin^{2} A = 1 - 2 sin^{2} A = 2 cos^{2} A - 1$
$tan 2 A = \frac{2 tan A}{1 - tan ^{2} A},$ $cos 2 A = \frac{1 - tan ^{2} A}{1 + tan ^{2} A}$
Triple angle
$sin 3 A = 3 sin A - 4 sin^{3} A,$ $cos 3 A = 4 cos^{3} A - 3 cos A,$ $tan 3 A = \frac{3 tan A - tan ^{3} A}{1 - 3 tan ^{2} A}$
Half angle
$sin^{2} \frac{A}{2} = \frac{1 - cos A}{2},$ $cos^{2} \frac{A}{2} = \frac{1 + cos A}{2},$ $tan \frac{A}{2} = \frac{1 - cos A}{sin A}$

Sum to product
$sin C + sin D = 2 sin \frac{C + D}{2} cos \frac{C - D}{2},$ $cos C + cos D = 2 cos \frac{C + D}{2} cos \frac{C - D}{2}$
$sin C - sin D = 2 cos \frac{C + D}{2} sin \frac{C - D}{2},$ $cos C - cos D = - 2 sin \frac{C + D}{2} sin \frac{C - D}{2}$
Product to sum
$2 sin A cos B = sin (A + B) + sin (A - B),$ $2 cos A cos B = cos (A + B) + cos (A - B),$ $2 sin A sin B = cos (A - B) - cos (A + B)$

Values
$sin 1 5^{\circ} = \frac{3 - 1}{2 2},$ $tan 1 5^{\circ} = 2 - 3,$ $sin 1 8^{\circ} = \frac{5 - 1}{4},$ $cos 3 6^{\circ} = \frac{5 + 1}{4}$
Product of cosines
$cos A cos 2 A cos 4 A \dots cos 2^{n - 1} A = \frac{sin 2 ^{n} A}{2 ^{n} sin A}$
Sines in AP
$k = 0 \sum n - 1 sin (α + k β) = \frac{sin \frac{n β}{2}}{sin \frac{β}{2}} sin (α + \frac{( n - 1 ) β}{2})$
Same for cosines with $cos$ in place of the last $sin$ .
Three-angle product
$sin θ sin (6 0^{\circ} - θ) sin (6 0^{\circ} + θ) = \frac{1}{4} sin 3 θ$