Trigonometry

Pythagorean identities:

• sin²θ + cos²θ = 1
• 1 + tan²θ = sec²θ
• 1 + cot²θ = csc²θ

Angle sum / difference:

• sin(A ± B) = sin A cos B ± cos A sin B
• cos(A ± B) = cos A cos B ∓ sin A sin B
• tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

Double angle:

• sin 2A = 2 sin A cos A
• cos 2A = cos²A − sin²A = 1 − 2sin²A = 2cos²A − 1
• tan 2A = 2 tan A / (1 − tan²A)

Half angle:

• sin(A/2) = ±√((1 − cos A)/2)
• cos(A/2) = ±√((1 + cos A)/2)
• tan(A/2) = sin A / (1 + cos A) = (1 − cos A) / sin A

Sum-to-product:

• sin C + sin D = 2 sin((C+D)/2) cos((C−D)/2)
• sin C − sin D = 2 cos((C+D)/2) sin((C−D)/2)
• cos C + cos D = 2 cos((C+D)/2) cos((C−D)/2)
• cos C − cos D = −2 sin((C+D)/2) sin((C−D)/2)

Product-to-sum:

• 2 sin A cos B = sin(A+B) + sin(A−B)
• 2 cos A sin 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)

Triple angle:

• sin 3A = 3 sin A − 4 sin³A
• cos 3A = 4 cos³A − 3 cos A
• tan 3A = (3 tan A − tan³A) / (1 − 3 tan²A)