Study Notes on Verifying Trigonometric Identities

Chapter 5.2 - Verifying Trigonometric Identities

Introduction to Verifying Trigonometric Identities

  • To verify a given trigonometric identity, the goal is to show that each side simplifies to the same value.
  • The process involves transforming the more complicated side of the equation into the form of the simpler side.
  • Important Note: You can only work with one side of the equation during verification.

Examples of Verifying Trigonometric Identities

Example 1: Verification Exercises

a) Verify that:

\tan A = \frac{\cos A}{\sin A} \cdot \cot^2 A

  • This utilizes the quotient identity.
b) Verify that:

\tan x \cos x = 1 - \cos^2 x

  • This involves applying the quotient and identity:
  • $\tan x = \frac{\sin x}{\cos x}$, then simplifying.
c) Verify that:

\csc A \sec A = \cot A + \tan A

  • This uses the identity relationships of cosecant and secant in relation to cotangent and tangent.
d) Verify that:

(1 - \cos 0)(1 + \sec 0) \cot 0 = \sin 0

  • Apply the quotient identity and reciprocal identity.
  • Combine using a common denominator:
    \frac{\sin^2 A}{\cos^2 A} .
e) Verify that:

\sin A \cos A (\tan A + \cot A) = 1

  • Utilize sine and cosine identities; simplify to show both sides equal 1.
f) Verify that:

1 - \sin^2 A = \sin A

  • This simplifies according to the Pythagorean identity:
    1 - \sin^2 A = \cos^2 A .
g) Verify that:

\frac{1}{1 - \sin A} \cdot \frac{1}{1 + \sin A} = 2 \sec^2 A

  • Combine both fractions over a common denominator, simplify to identify secant squared.
h) Verify that:

\tan + \sin + \cos 0 = \sec

  • Use the quotient identity and simplify using sine and cosine relationships.
i) Verify that:

(1 - \sin A)(1 + \sin A) = 2 \sec^2 A

  • Expanding shows Pythagorean relations.
j) Verify that:

1 + \sec x = \sin x + \cos x \cdot \tan x + \cot x

  • Evaluate common denominators to confirm equality.

Additional Example - Non-Identity Explanation

Example 2:

  • Analyze why the following equation is not an identity:
    \tan 0 = \sec^2 x - 1
  • Recognize this employs the Pythagorean identity:
    \sec^2 x - 1 = \tan^2 x
  • The left side may yield negative values, whereas the right side must always be positive; thus, the equation does not hold true for all values of 0.

Homework Assignments

  • Page 362 - 363: Problems #13 - 45 (odd), 59, 61

End of Notes