trigonometric identities

• Reciprocal identities include sin 𝜃 = 1 csc 𝜃, cos 𝜃 = 1 sec 𝜃, and tan 𝜃 = 1 cot 𝜃.
• Ratio identities include tan 𝜃 = sin �� cos 𝜃, tan �� = sec �� csc 𝜃, and cot �� = cos 𝜃 sin 𝜃.
• Pythagorean identities include sin 2 𝜃 + cos 2 𝜃 = 1, tan 2 𝜃 + 1 = sec 2 𝜃, and 1 + cot 2 𝜃 = csc 2 𝜃.
• Identities due to symmetry include sin(−��) = − sin 𝜃, cos(−𝜃) = cos 𝜃, and tan(−𝜃) = − tan 𝜃.
• Sum and Difference Identities include cos(�� + 𝛽) = cos 𝛼 cos �� − sin 𝛼 sin ��, cos(𝛼 − 𝛽) = cos 𝛼 cos 𝛽 + sin 𝛼 sin 𝛽, sin(𝛼 + 𝛽) = sin 𝛼 cos 𝛽 + cos 𝛼 sin 𝛽, and sin(𝛼 − 𝛽) = sin 𝛼 cos 𝛽 − cos 𝛼 sin 𝛽.
• Cofunction Identities include cos (𝜋 2 − 𝑡) = sin 𝑡 (in radians), cos(90 − 𝑡) = sin 𝑡 (in degrees), sin (𝜋 2 − 𝑡) = cos 𝑡 (in radians), and sin(90 − 𝑡) = cos 𝑡 (in degrees).
• Double-Angle Identities include cos 2�� = cos 2 𝛼 − sin 2 𝛼 = 1 − 2 sin 2 𝛼, sin(2��) = 2 sin 𝛼 cos ��, and tan(2𝛼) = 2 tan 𝛼 1−tan 2 𝛼.
• Power Reduction Identities include cos 2 𝛼 = 1+cos(2𝛼) 2, sin 2 𝛼 = 1−cos(2𝛼) 2, and tan 2 𝛼 = 1−cos(2𝛼) 1+cos(2𝛼) .
• Half-Angle Identities include cos (𝑢 2 ) = ±√ 1+cos 𝑢 2, sin (𝑢 2 ) = ±√ 1−cos 𝑢 2, and tan (𝑢 2 ) = ±√ 1−cos 𝑢 1+cos 𝑢 = 1−cos 𝑢 sin 𝑢.