Trigonometric identities

Trigonometric identities are equalities that involve trigonometric functions and are true for every single value of the occurring variables. The following are the basic trigonometric identities and are true for all angels except those for which either side of the equation is undefined:

$$cos^{2}\Theta +sin^{2}\Theta =1$$

$$tan^{2}\Theta +1=sec^{2}\Theta$$

$$cot^{2}\Theta +1=csc^{2}\Theta$$

The following trigonometric identities are called the sum and difference of angles formulas

$$cos(\alpha + \beta) =cos\: \alpha \: cos\: \beta - sin\: \alpha \: sin\: \beta$$

$$cos(\alpha- \beta) =cos\: \alpha \: cos\: \beta + sin\: \alpha \: sin\: \beta$$

$$sin(\alpha + \beta) =sin\: \alpha \: cos\: \beta + cos\: \alpha \: sin\: \beta$$

$$sin(\alpha - \beta) =sin\: \alpha \: cos\: \beta - cos\: \alpha \: sin\: \beta$$

The following trigonometric identities are called the double-angle formulas

$$\\ sin\: 2\Theta =2sin\Theta cos\Theta \\ cos\: 2\Theta =cos^{2}\Theta -sin^{2}\Theta \\ cos\: 2\Theta =1 -2sin^{2}\Theta \\ cos\: 2\Theta =2cos^{2}\Theta -1\\$$

The following trigonometric identities are called the half-angle formulas

$$cos\frac{\Theta }{2}=\pm \sqrt{\frac{1+cos\: \Theta }{2}}$$

$$sin\frac{\Theta }{2}=\pm \sqrt{\frac{1-cos\: \Theta }{2}}$$


Video lesson

Show that the following statement must be true 1+cos 2Ѳ= 2cos2Ѳ