Trig Identites

Reciprocal Identities

  1. sin⁡θ=1csc⁡θ\sin \theta = \frac{1}{\csc \theta}sinθ=cscθ1​

  2. cos⁡θ=1sec⁡θ\cos \theta = \frac{1}{\sec \theta}cosθ=secθ1​

  3. tan⁡θ=1cot⁡θ\tan \theta = \frac{1}{\cot \theta}tanθ=cotθ1​

  4. csc⁡θ=1sin⁡θ\csc \theta = \frac{1}{\sin \theta}cscθ=sinθ1​

  5. sec⁡θ=1cos⁡θ\sec \theta = \frac{1}{\cos \theta}secθ=cosθ1​

  6. cot⁡θ=1tan⁡θ\cot \theta = \frac{1}{\tan \theta}cotθ=tanθ1​

Pythagorean Identities

  1. sin⁡2θ+cos⁡2θ=1\sin^2 \theta + \cos^2 \theta = 1sin2θ+cos2θ=1

  2. 1+tan⁡2θ=sec⁡2θ1 + \tan^2 \theta = \sec^2 \theta1+tan2θ=sec2θ

  3. 1+cot⁡2θ=csc⁡2θ1 + \cot^2 \theta = \csc^2 \theta1+cot2θ=csc2θ

Quotient Identities

  1. tan⁡θ=sin⁡θcos⁡θ\tan \theta = \frac{\sin \theta}{\cos \theta}tanθ=cosθsinθ​

  2. cot⁡θ=cos⁡θsin⁡θ\cot \theta = \frac{\cos \theta}{\sin \theta}cotθ=sinθcosθ​

Co-Function Identities

  1. sin⁡(90∘−θ)=cos⁡θ\sin(90^\circ - \theta) = \cos \thetasin(90∘−θ)=cosθ

  2. cos⁡(90∘−θ)=sin⁡θ\cos(90^\circ - \theta) = \sin \thetacos(90∘−θ)=sinθ

  3. tan⁡(90∘−θ)=cot⁡θ\tan(90^\circ - \theta) = \cot \thetatan(90∘−θ)=cotθ

  4. cot⁡(90∘−θ)=tan⁡θ\cot(90^\circ - \theta) = \tan \thetacot(90∘−θ)=tanθ

  5. sec⁡(90∘−θ)=csc⁡θ\sec(90^\circ - \theta) = \csc \thetasec(90∘−θ)=cscθ

  6. csc⁡(90∘−θ)=sec⁡θ\csc(90^\circ - \theta) = \sec \thetacsc(90∘−θ)=secθ

Even-Odd Identities

  • Even Functions (symmetric about the y-axis):

    1. cos⁡(−θ)=cos⁡θ\cos(-\theta) = \cos \thetacos(−θ)=cosθ

    2. sec⁡(−θ)=sec⁡θ\sec(-\theta) = \sec \thetasec(−θ)=secθ

  • Odd Functions (symmetric about the origin):
    3. sin⁡(−θ)=−sin⁡θ\sin(-\theta) = -\sin \thetasin(−θ)=−sinθ
    4. tan⁡(−θ)=−tan⁡θ\tan(-\theta) = -\tan \thetatan(−θ)=−tanθ
    5. csc⁡(−θ)=−csc⁡θ\csc(-\theta) = -\csc \thetacsc(−θ)=−cscθ
    6. cot⁡(−θ)=−cot⁡θ\cot(-\theta) = -\cot \thetacot(−θ)=−cotθ

Sum and Difference Formulas

  1. Sine:

    • sin⁡(a+b)=sin⁡acos⁡b+cos⁡asin⁡b\sin(a + b) = \sin a \cos b + \cos a \sin bsin(a+b)=sinacosb+cosasinb

    • sin⁡(a−b)=sin⁡acos⁡b−cos⁡asin⁡b\sin(a - b) = \sin a \cos b - \cos a \sin bsin(a−b)=sinacosb−cosasinb

  2. Cosine:

    • cos⁡(a+b)=cos⁡acos⁡b−sin⁡asin⁡b\cos(a + b) = \cos a \cos b - \sin a \sin bcos(a+b)=cosacosb−sinasinb

    • cos⁡(a−b)=cos⁡acos⁡b+sin⁡asin⁡b\cos(a - b) = \cos a \cos b + \sin a \sin bcos(a−b)=cosacosb+sinasinb

  3. Tangent:

    • tan⁡(a+b)=tan⁡a+tan⁡b1−tan⁡atan⁡b\tan(a + b) = \frac{\tan a + \tan b}{1 - \tan a \tan b}tan(a+b)=1−tanatanbtana+tanb​

    • tan⁡(a−b)=tan⁡a−tan⁡b1+tan⁡atan⁡b\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}tan(a−b)=1+tanatanbtana−tanb​

Double Angle Formulas

  1. Sine: sin⁡2θ=2sin⁡θcos⁡θ\sin 2\theta = 2\sin \theta \cos \thetasin2θ=2sinθcosθ

  2. Cosine: cos⁡2θ=cos⁡2θ−sin⁡2θ\cos 2\theta = \cos^2 \theta - \sin^2 \thetacos2θ=cos2θ−sin2θ Alternative forms: cos⁡2θ=2cos⁡2θ−1\cos 2\theta = 2\cos^2 \theta - 1cos2θ=2cos2θ−1 cos⁡2θ=1−2sin⁡2θ\cos 2\theta = 1 - 2\sin^2 \thetacos2θ=1−2sin2θ

  3. Tangent: tan⁡2θ=2tan⁡θ1−tan⁡2θ\tan 2\theta = \frac{2\tan \theta}{1 - \tan^2 \theta}tan2θ=1−tan2θ2tanθ​

Half-Angle Formulas

  1. Sine: sin⁡θ2=±1−cos⁡θ2\sin \frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos \theta}{2}}sin2θ​=±21−cosθ​​

  2. Cosine: cos⁡θ2=±1+cos⁡θ2\cos \frac{\theta}{2} = \pm\sqrt{\frac{1 + \cos \theta}{2}}cos2θ​=±21+cosθ​​

  3. Tangent: tan⁡θ2=±1−cos⁡θ1+cos⁡θ\tan \frac{\theta}{2} = \pm\sqrt{\frac{1 - \cos \theta}{1 + \cos \theta}}tan2θ​=±1+cosθ1−cosθ​​ Alternative forms: tan⁡θ2=sin⁡θ1+cos⁡θ=1−cos⁡θsin⁡θ\tan \frac{\theta}{2} = \frac{\sin \theta}{1 + \cos \theta} = \frac{1 - \cos \theta}{\sin \theta}tan2θ​=1+cosθsinθ​=sinθ1−cosθ​