Trigonometric identities Knowledge-Based Study

Study notes

1. Reciprocal and Quotient Identities

   Trigonometric identities are mathematical equations involving trigonometric functions that are valid for all values within their shared domains.

   The foundational relations link the primary trigonometric functions (sine, cosine, and tangent) with their reciprocals:

   • Reciprocal Identities:
     • $\csc(\theta) = \frac{1}{\sin(\theta)}$ for $\sin(\theta) \neq 0$
     • $\sec(\theta) = \frac{1}{\cos(\theta)}$ for $\cos(\theta) \neq 0$
     • $\cot(\theta) = \frac{1}{\tan(\theta)}$ for $\tan(\theta) \neq 0$
   • Quotient Identities:
     • $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}$ for $\cos(\theta) \neq 0$
     • $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$ for $\sin(\theta) \neq 0$

   The Pythagorean Identities

   The primary Pythagorean identity arises directly from the unit circle equation $x^2 + y^2 = 1$, where $x = \cos(\theta)$ and $y = \sin(\theta)$:

   $$\sin^2(\theta) + \cos^2(\theta) = 1$$

   Two auxiliary identities are derived by dividing this core equation by either $\cos^2(\theta)$ or $\sin^2(\theta)$:

   • Dividing by $\cos^2(\theta)$:
     $\frac{\sin^2(\theta)}{\cos^2(\theta)} + \frac{\cos^2(\theta)}{\cos^2(\theta)} = \frac{1}{\cos^2(\theta)} \implies 1 + \tan^2(\theta) = \sec^2(\theta)$
   • Dividing by $\sin^2(\theta)$:
     $\frac{\sin^2(\theta)}{\sin^2(\theta)} + \frac{\cos^2(\theta)}{\sin^2(\theta)} = \frac{1}{\sin^2(\theta)} \implies 1 + \cot^2(\theta) = \csc^2(\theta)$

   Rearranged forms, such as $\sin^2(\theta) = 1 - \cos^2(\theta)$ or $\sec^2(\theta) - \tan^2(\theta) = 1$, are frequently used to substitute terms in algebraic expressions.

   Techniques for Simplification

   Simplifying a trigonometric expression means rewriting it in a more concise form, often involving a single trigonometric function or numerical constant.

   Key strategies include:

   1. Converting to Sine and Cosine: Express all instances of $\tan(\theta)$, $\cot(\theta)$, $\sec(\theta)$, and $\csc(\theta)$ in terms of $\sin(\theta)$ and $\cos(\theta)$.
   2. Combining Fractions: Find common denominators to merge separate rational terms.
   3. Factoring: Factor out greatest common factors or recognize patterns like differences of squares $a^2 - b^2 = (a - b)(a + b)$.
   4. Pythagorean Substitution: Replace squared terms using Pythagorean identities.

   Worked Example: Simplify the expression $\frac{\cos(\theta)}{1 - \sin(\theta)} - \tan(\theta)$.

   Rewrite $\tan(\theta)$ using the quotient identity:

   $$\frac{\cos(\theta)}{1 - \sin(\theta)} - \frac{\sin(\theta)}{\cos(\theta)}$$

   Find a common denominator, $\cos(\theta)(1 - \sin(\theta))$:

   $$\frac{\cos^2(\theta) - \sin(\theta)(1 - \sin(\theta))}{\cos(\theta)(1 - \sin(\theta))} = \frac{\cos^2(\theta) - \sin(\theta) + \sin^2(\theta)}{\cos(\theta)(1 - \sin(\theta))}$$

   Apply $\sin^2(\theta) + \cos^2(\theta) = 1$:

   $$\frac{1 - \sin(\theta)}{\cos(\theta)(1 - \sin(\theta))} = \frac{1}{\cos(\theta)} = \sec(\theta)$$

   Verifying Identities

   Verifying an identity requires showing that one side of an equation can be transformed into the other side using established mathematical rules.

   Standard guidelines for identity verification:

   • Start with the more complicated side of the identity.
   • Work on only one side at a time; do not treat the statement as an equation by adding, subtracting, multiplying, or dividing both sides.
   • Keep the target side in view to guide your choice of substitutions.

   Worked Example: Verify that $\frac{1}{1 - \cos(x)} + \frac{1}{1 + \cos(x)} = 2\csc^2(x)$.

   Begin with the left-hand side (LHS):

   $$\text{LHS} = \frac{(1 + \cos(x)) + (1 - \cos(x))}{(1 - \cos(x))(1 + \cos(x))}$$

   Combine terms in the numerator and expand the denominator:

   $$\text{LHS} = \frac{2}{1 - \cos^2(x)}$$

   Substitute $1 - \cos^2(x) = \sin^2(x)$:

   $$\text{LHS} = \frac{2}{\sin^2(x)} = 2\csc^2(x) = \text{RHS}$$

   Since the left-hand side simplifies directly to the right-hand side, the identity is verified.