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Guidelines for Verifying Trigonometric Identities: Step-by-Step Approach

This guide outlines key techniques for verifying trigonometric identities effectively. Focus on one side of the equation, ideally the more complex side, to simplify your work. Explore opportunities for factoring, adding fractions, or converting terms to sines and cosines using fundamental identities. Always attempt different strategies, as even failed attempts can provide valuable insights. Remember, you cannot assume the two sides are equal and certain operations like adding quantities to both sides are not permissible.

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Guidelines for Verifying Trigonometric Identities: Step-by-Step Approach

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  1. Section 5.2 Verifying Trigonometric Identities

  2. Guidelines for Verifying Trigonometric Identities

  3. 1. Work with one side of the equation. It is often better to work with the more complicated side. • 2. Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator.

  4. 3. Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents. • 4. If the preceding guidelines do not help, try converting all terms to sines and cosines. • 5. ALWAYS TRY SOMETHING!!! Even paths that lead to dead ends provide insights.

  5. When verifying an identity you cannot assume that the two sides of the equation are equal because you are trying to verify that they are equal. • As a result, when verifying identities, you cannot use operations such as adding the same quantity to each side of the equation or cross multiplication.

  6. Example 1 • Verify the identity.

  7. tan4x = tan2x sec2x − tan2x • (tan2x)(tan2x) = tan2x sec2x − tan2x • (tan2x)(sec2x − 1) = tan2x sec2x − tan2x • tan2x sec2x − tan2x = tan2x sec2x − tan2x

  8. sin3x cos4x = (cos4x − cos6x) sinx • sin2x cos4x sin x = (cos4x − cos6x) sinx • (1 − cos2x)cos4x sin x = (cos4x − cos6x) sinx • (cos4x − cos6x) sinx = (cos4x − cos6x) sinx

  9. csc4x cotx = csc2x(cot x + cot3x) • csc2xcsc2x cotx = csc2x(cot x + cot3x) • csc2x(1 + cot2x)cot x = csc2x(cot x + cot3x) • csc2x(cot x + cot3x) = csc2x(cot x + cot3x)

  10. Never work with both sides of an identity. • Pick one side and work with that side.

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