# Trigonometric Identities

I. Introduction: Trigonometric identities are equalities involving trigonometric functions that are true for all values of the variables within their domains. Understanding these identities is essential for simplifying expressions, solving equations, and proving mathematical relationships.

II. Pythagorean Identities:

1. Pythagorean Identity:

• ${\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta =1$
• This fundamental identity holds for all values of $\theta$ and is derived from the Pythagorean theorem in a right-angled triangle.
2. Reciprocal Pythagorean Identities:

• ${\mathrm{csc}}^{2}\theta =1+{\mathrm{cot}}^{2}\theta$
• ${\mathrm{sec}}^{2}\theta =1+{\mathrm{tan}}^{2}\theta$
• ${\mathrm{cot}}^{2}\theta ={\mathrm{csc}}^{2}\theta -1$
• ${\mathrm{tan}}^{2}\theta ={\mathrm{sec}}^{2}\theta -1$

III. Reciprocal Identities:

1. Cosecant ($\mathrm{csc}$) Reciprocal Identity:

• $\mathrm{csc}\theta =\frac{1}{\mathrm{sin}\theta }$
2. Secant ($\mathrm{sec}$) Reciprocal Identity:

• $\mathrm{sec}\theta =\frac{1}{\mathrm{cos}\theta }$
3. Cotangent ($\mathrm{cot}$) Reciprocal Identity:

• $\mathrm{cot}\theta =\frac{1}{\mathrm{tan}\theta }$

IV. Quotient and Co-Function Identities:

1. Tangent ($\mathrm{tan}$) Quotient Identity:

• $\mathrm{tan}\theta =\frac{\mathrm{sin}\theta }{\mathrm{cos}\theta }$
2. Cotangent ($\mathrm{cot}$) Quotient Identity:

• $\mathrm{cot}\theta =\frac{\mathrm{cos}\theta }{\mathrm{sin}\theta }$
3. Secant ($\mathrm{sec}$) and Cosecant ($\mathrm{csc}$) Co-Function Identities:

• $\mathrm{sec}\theta =\frac{1}{\mathrm{cos}\theta }$
• $\mathrm{csc}\theta =\frac{1}{\mathrm{sin}\theta }$

V. Double Angle Formulas:

1. $\mathrm{sin}\left(2\theta \right)=2\mathrm{sin}\theta \mathrm{cos}\theta$
2. $\mathrm{cos}\left(2\theta \right)={\mathrm{cos}}^{2}\theta -{\mathrm{sin}}^{2}\theta$
3. $\mathrm{tan}\left(2\theta \right)=\frac{2\mathrm{tan}\theta }{1-{\mathrm{tan}}^{2}\theta }$

VI. Half-Angle Formulas:

1. $\mathrm{sin}\left(\frac{\theta }{2}\right)=±\sqrt{\frac{1-\mathrm{cos}\theta }{2}}$
2. $\mathrm{cos}\left(\frac{\theta }{2}\right)=±\sqrt{\frac{1+\mathrm{cos}\theta }{2}}$
3. $\mathrm{tan}\left(\frac{\theta }{2}\right)=±\sqrt{\frac{1-\mathrm{cos}\theta }{1+\mathrm{cos}\theta }}$

VII. Sum and Difference Formulas:

1. $\mathrm{sin}\left(A+B\right)=\mathrm{sin}A\mathrm{cos}B+\mathrm{cos}A\mathrm{sin}B$
2. $\mathrm{cos}\left(A+B\right)=\mathrm{cos}A\mathrm{cos}B-\mathrm{sin}A\mathrm{sin}B$
3. $\mathrm{tan}\left(A+B\right)=\frac{\mathrm{tan}A+\mathrm{tan}B}{1-\mathrm{tan}A\mathrm{tan}B}$