Trigonometric identities are like secret codes for angles. They help us simplify complex trig expressions and solve tricky equations. These formulas connect different trig functions, making it easier to work with angles in all sorts of math problems.

Knowing these identities is crucial for tackling advanced trig problems. They're the building blocks for proving other trig relationships and solving equations. Mastering them will give you a solid foundation for more complex math concepts down the road.

Trigonometric Identities

Fundamental Trigonometric Identities

• The Pythagorean identity states that for any angle θ, $sin^2θ + cos^2θ = 1$. This identity finds the value of the sine, cosine, or tangent of an angle given one of the other two values
• The reciprocal identities define the secant, cosecant, and cotangent functions in terms of sine, cosine, and tangent: $sec θ = 1/cos θ$, $csc θ = 1/sin θ$, and $cot θ = 1/tan θ = cos θ/sin θ$
• The quotient identities express tangent and cotangent in terms of sine and cosine: $tan θ = sin θ/cos θ$ and $cot θ = cos θ/sin θ$

Identities Involving Angle Relationships

• The co-function identities relate the trigonometric functions of complementary angles: $sin(90° - θ) = cos θ$, $cos(90° - θ) = sin θ$, $tan(90° - θ) = cot θ$, $cot(90° - θ) = tan θ$, $sec(90° - θ) = csc θ$, and $csc(90° - θ) = sec θ$
• The odd-even identities describe the behavior of the trigonometric functions under changes in sign of the angle: $cos(-θ) = cos θ$, $sin(-θ) = -sin θ$, $tan(-θ) = -tan θ$, $cot(-θ) = -cot θ$, $sec(-θ) = sec θ$, and $csc(-θ) = -csc θ$
• The sum and difference identities for sine, cosine, and tangent find the exact values of trigonometric functions for angles that are the sum or difference of two angles with known trigonometric values ($sin(α ± β)$, $cos(α ± β)$, $tan(α ± β)$)

Simplifying Trigonometric Expressions

Applying Fundamental Identities

• Trigonometric expressions can be simplified by applying the fundamental identities, such as the Pythagorean, reciprocal, quotient, co-function, odd-even, sum, and difference identities
• Simplifying trigonometric expressions often involves rewriting the expression in terms of sine and cosine, using the reciprocal and quotient identities ($sec θ = 1/cos θ$, $csc θ = 1/sin θ$, $tan θ = sin θ/cos θ$, $cot θ = cos θ/sin θ$)

Simplifying Expressions with Double and Half Angles

• The double-angle identities express the sine, cosine, and tangent of double angles $(2θ)$ in terms of the sine and cosine of the original angle $(θ)$. These identities simplify expressions containing double angles ($sin 2θ$, $cos 2θ$, $tan 2θ$)
• The half-angle identities express the sine, cosine, and tangent of half angles $(θ/2)$ in terms of the sine and cosine of the original angle $(θ)$. These identities simplify expressions containing half angles ($sin (θ/2)$, $cos (θ/2)$, $tan (θ/2)$)
• The power-reducing identities, such as $sin^2θ = (1 - cos 2θ)/2$ and $cos^2θ = (1 + cos 2θ)/2$, simplify expressions with higher powers of sine and cosine

Verifying Trigonometric Identities

Steps for Verifying Identities

• To verify a trigonometric identity, start with the more complex side of the equation and simplify it using known identities until it matches the other side of the equation
• Verifying identities often involves rewriting expressions in terms of sine and cosine using the reciprocal and quotient identities ($sec θ = 1/cos θ$, $csc θ = 1/sin θ$, $tan θ = sin θ/cos θ$, $cot θ = cos θ/sin θ$)
• When verifying identities, ensure that each step is reversible and that the domain of the expressions remains the same throughout the process

Graphical Verification of Identities

• Trigonometric identities can also be verified graphically by plotting both sides of the equation and observing that the graphs are identical for all values in the domain
• Graphical verification provides a visual confirmation of the identity and helps understand the relationship between the expressions on both sides of the equation

Solving Trigonometric Equations

Applying Identities to Solve Equations

• Trigonometric equations can be solved by applying identities to simplify the equation and then using algebraic techniques to isolate the variable
• When solving trigonometric equations, consider the domain of the functions involved and the periodicity of the solutions
• The Pythagorean identity ($sin^2θ + cos^2θ = 1$) solves equations involving multiple trigonometric functions
• The sum and difference identities ($sin(α ± β)$, $cos(α ± β)$, $tan(α ± β)$) solve equations involving trigonometric functions of the sum or difference of angles
• The double-angle and half-angle identities ($sin 2θ$, $cos 2θ$, $tan 2θ$, $sin (θ/2)$, $cos (θ/2)$, $tan (θ/2)$) solve equations involving trigonometric functions of double or half angles

Checking Solutions

• After solving a trigonometric equation, check the solutions by substituting them back into the original equation to ensure that they satisfy the equation for all values in the domain
• Checking solutions helps identify any extraneous solutions that may have been introduced during the solving process and ensures the accuracy of the final answer