Trig Identities Calculator
Verify and understand various trigonometric identities.
Trigonometric Identities
Pythagorean Identities
- \[ \sin^2(x) + \cos^2(x) = 1 \]
- \[ \tan^2(x) + 1 = \sec^2(x) \]
- \[ \cot^2(x) + 1 = \csc^2(x) \]
Reciprocal Identities
- \[ \sin(x) = \frac{1}{\csc(x)} \]
- \[ \cos(x) = \frac{1}{\sec(x)} \]
- \[ \tan(x) = \frac{1}{\cot(x)} \]
Quotient Identities
- \[ \tan(x) = \frac{\sin(x)}{\cos(x)} \]
- \[ \cot(x) = \frac{\cos(x)}{\sin(x)} \]
Cofunction Identities
- \[ \sin(x) = \cos(90° - x) \]
- \[ \cos(x) = \sin(90° - x) \]
- \[ \tan(x) = \cot(90° - x) \]
- \[ \cot(x) = \tan(90° - x) \]
- \[ \sec(x) = \csc(90° - x) \]
- \[ \csc(x) = \sec(90° - x) \]
Even-Odd Identities
- \[ \sin(-x) = -\sin(x) \]
- \[ \cos(-x) = \cos(x) \]
- \[ \tan(-x) = -\tan(x) \]
Double Angle Identities
- \[ \sin(2x) = 2\sin(x)\cos(x) \]
- \[ \cos(2x) = \cos^2(x) - \sin^2(x) \]
- \[ \tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)} \]
Half Angle Identities
- \[ \sin(\frac{x}{2}) = \pm\sqrt{\frac{1 - \cos(x)}{2}} \]
- \[ \cos(\frac{x}{2}) = \pm\sqrt{\frac{1 + \cos(x)}{2}} \]
- \[ \tan(\frac{x}{2}) = \frac{\sin(x)}{1 + \cos(x)} \]
Sum and Difference Identities
- \[ \sin(A \pm B) = \sin(A)\cos(B) \pm \cos(A)\sin(B) \]
- \[ \cos(A \pm B) = \cos(A)\cos(B) \mp \sin(A)\sin(B) \]
- \[ \tan(A \pm B) = \frac{\tan(A) \pm \tan(B)}{1 \mp \tan(A)\tan(B)} \]
Product-to-Sum Identities
- \[ \sin(A)\sin(B) = \frac{1}{2}[\cos(A-B) - \cos(A+B)] \]
- \[ \cos(A)\cos(B) = \frac{1}{2}[\cos(A-B) + \cos(A+B)] \]
- \[ \sin(A)\cos(B) = \frac{1}{2}[\sin(A+B) + \sin(A-B)] \]
Sum-to-Product Identities
- \[ \sin(A) + \sin(B) = 2\sin(\frac{A+B}{2})\cos(\frac{A-B}{2}) \]
- \[ \sin(A) - \sin(B) = 2\cos(\frac{A+B}{2})\sin(\frac{A-B}{2}) \]
- \[ \cos(A) + \cos(B) = 2\cos(\frac{A+B}{2})\cos(\frac{A-B}{2}) \]
- \[ \cos(A) - \cos(B) = -2\sin(\frac{A+B}{2})\sin(\frac{A-B}{2}) \]