Chapter 10: Trig Identities and Proving Identities

Graphs and Transformations

• The discussion centers on recognizing a graph that most closely resembles a tangent-shaped graph.
• Goal: identify a formula in terms of tangent, and note that shifting it left or right complicates things.
• Vertical asymptotes:
  • Tangent does not have a vertical asymptote at zero and essentially straddles the y-axis.
  • Cotangent is the other standard trig function with vertical asymptotes (at multiples of
    x = k\,\pi).
• The graph in question is not exactly cotangent; it’s flipped relative to cotangent.
  • A simple fix is to apply a minus sign in front (i.e., multiply by
    -1), which is often easier than horizontal shifts.
• Upside-down form: use a negative out front to reflect the graph.
• Period considerations:
  • The normal period of tangent is \pi.
  • The graph being discussed appears to have a period of 2\pi until a stretch is determined.
  • The speaker leaves the period blanks for now, indicating uncertainty about stretch and exact period.
• Vertical/vertical-logical shift: the graph needs to be shifted up by 1 unit (e.g., from starting points like (-1, 0, 1) to (0, 1, 2) in the relevant y-values).
• Summary takeaway: identify a tangent-like transformation, use a negative sign if flipped, and consider vertical shifts before fixing the period.

Chapter 10: Identities and Proving Identities

• Chapter focus: developing methods to prove identities and use these identities to evaluate expressions not on the unit circle (e.g., 15° values).
• Foundational practice: memorize the identities already discussed; they will appear on the formula sheet.
• Core guidance: you should know these identities by heart and be able to recall them quickly today.
• Important caveat: there will be additional identities provided on the formula sheet, but memorization remains crucial for quick recall during problem solving.

What an identity is

• An identity is an equation that is true for every value in the domain (i.e., for all angles).
• Notation: angles can be represented with different symbols (a, b, θ, x, etc.), but the identity must hold for all of them.
• Distinction:
  • Prove an identity: show the equality holds for all valid angles.
  • Verify an identity: if the problem states 'verify', you may treat the identity as true and show the left-hand side equals the right-hand side.
• Strategy emphasis: start with the side that looks messier; manipulate algebraically to reveal equivalence.
• Example idea: use algebraic moves (multiplication/division by same nonzero quantity, factoring, expanding) to reveal equality, while ensuring you can justify each step.

Key trig identities (recap and use in proofs)

• Reciprocal identities:
  • Tangent and cotangent relationship:
    \cot x = \frac{1}{\tan x}
  • (Note: other reciprocal identities for sine/cosine include csc x = 1/ sin x and sec x = 1/ cos x, though only cot x is explicitly mentioned here.)
• Pythagorean identities:
  • Basic:
    \sin^{2}x + \cos^{2}x = 1
  • Derived:
    \tan^{2}x + 1 = \sec^{2}x
  • Derivation hint: divide the left-hand side of the basic identity by \cos^{2}x to obtain the derived form.
• Using signs in quadrants (sign conventions): some identities depend on the quadrant, e.g., signs of sine/c cosine; ensure the sign is consistent when substituting.
• Expressing sine in terms of cosine (and vice versa):
  • From the basic identity:
    \sin^{2}x = 1 - \cos^{2}x
  • Consequently,
    \sin x = \pm \sqrt{1 - \cos^{2}x}
  • It's common to rewrite expressions entirely in terms of sine and cosine before simplifying.
• Practical note: when reducing complex expressions, replacing secant with its cosine form can simplify handling of quotients.

Proving identities: a practical workflow

• Step 1: Examine the left-hand side (LHS) and right-hand side (RHS) of the identity.
• Step 2: Look for a starting move that is obviously equivalent (e.g., replace secant with 1/ cos, or rewrite tangent as sine over cosine).
• Step 3: Simplify consistently, keeping track of how each operation transforms the expression.
• Step 4: If you get a messy quotient, consider rewriting everything in terms of sine and cosine and/or factor where useful.
• Step 5: If a path stalls, back up and try a different starting substitution. The key is to begin with something you know is equivalent and then proceed step by step so others can follow your logic.
• Step 6: The end goal is to show the left side is equal to the right side by a clear chain of equal expressions, with each step justified.

Worked approach to a sample identity (LHS: sec x sin x + cos x)

• Start with:
  \text{LHS} = \sec x \sin x + \cos x
• Replace secant:
  \text{LHS} = \frac{\sin x}{\cos x} + \cos x
• Get a common denominator:
  \text{LHS} = \frac{\sin x}{\cos x} + \frac{\cos^{2}x}{\cos x} = \frac{\sin x + \cos^{2}x}{\cos x}
• Express cos^2 x using the Pythagorean identity:
  \cos^{2}x = 1 - \sin^{2}x
• Substitute:
  \text{LHS} = \frac{\sin x + (1 - \sin^{2}x)}{\cos x} = \frac{1 + \sin x - \sin^{2}x}{\cos x}
• (Notes on further simplification): continuing from here would require either expressing everything in terms of sine or cosine, or using other identities to simplify the numerator and/or denominator. The key is to proceed step by step so the path is transparent to someone else following your work.
• Important practice point: do not skip steps; the goal is a transparent, reproducible chain of equalities.

Algebraic techniques relevant to trigonometric identities

• Factoring techniques (refer to algebra basics) that often appear in identity proofs:
  • Difference of squares:
    a^{2} - b^{2} = (a - b)(a + b)
  • Sum of cubes:
    a^{3} + b^{3} = (a + b)(a^{2} - ab + b^{2})
  • Difference of cubes:
    a^{3} - b^{3} = (a - b)(a^{2} + ab + b^{2})
• When a problem asks you to prove an identity, you may need to manipulate each side separately before showing they are equal; however, you should maintain the invariant that you are not introducing extraneous steps that would not hold for all angles.
• Strategy: start with a form you know is equivalent, then simplify until you reach a common expression or a form that clearly matches the other side.

Problem-solving strategies and learner tips

• Start with something you know is equivalent; if it doesn’t help, backtrack and try another route.
• The objective is to replace expressions with equivalent expressions and to show the chain of equivalences clearly to the reader.
• Remember: there can be multiple valid paths to the same identity; the key is to be transparent about each step.
• In exam settings, be prepared to show a complete, step-by-step derivation so a grader can follow your reasoning.

Quick algebra refresher you should know by heart

• Factor special quadratics (from algebra):
  • Difference of squares: a^2 - b^2 = (a - b)(a + b)
  • Sum of cubes: a^3 + b^3 = (a + b)(a^2 - ab + b^2)
  • Difference of cubes: a^3 - b^3 = (a - b)(a^2 + ab + b^2)
• Practice focus: recognize these patterns to simplify trigonometric expressions efficiently.

Off-topic note (digressions in the transcript)

• A segment of the transcript veered into casual conversations about video games (e.g., Ark Survival Ascended, Ark: Survival Evolved) and personal gaming discussions.
• These digressions are not part of the math content and should be ignored for study purposes.
• Some numeric chatter about hours played and platform specifics appeared; not relevant to the identities material.

Connections to broader topics and real-world relevance

• Identities underpin how we transform and simplify expressions in signal processing, physics, engineering, and computer graphics.
• Understanding period, symmetry, and shifts in trig graphs translates to waveform analysis and rotational motion problems.
• Mastery of algebraic factoring improves problem-solving speed across calculus (limits, derivatives) and linear algebra (eigenvalue proofs, transforms).

Quick reference formulas (recap)

• Reciprocal relation:
  \cot x = \frac{1}{\tan x}
• Pythagorean identities:
  \sin^{2}x + \cos^{2}x = 1\,,
  \tan^{2}x + 1 = \sec^{2}x\,.
• Basic sine-cosine identity for manipulation:
  \sin^{2}x = 1 - \cos^{2}x\,,
  \cos^{2}x = 1 - \sin^{2}x\,.
• Tangent and secant quotients:
  \tan x = \frac{\sin x}{\cos x}, \quad \sec x = \frac{1}{\cos x}, \quad \sec^{2}x = \frac{1}{\cos^{2}x}\,.
• Periods (for reference):
  • \text{period}(\tan x) = \pi
  • For a transformed or stretched graph, the period may be 2\pi or another value depending on the transformation applied.
• Typical transformation forms to recognize: shift, reflection, and dilation: e.g., \tan(x - h) (shift right by h), -\tan x (reflection about the x-axis).