Verify The Identity: Show Your Work On Cot Θ = Sec Θ = Csc

Verify the identity. Show your work. cot θ ∙ sec θ = csc θ 2.

The problem asks us to verify the identity: cot θ · sec θ = csc θ.

Recall the definitions of the trigonometric functions in terms of sine and cosine:

  • cot θ = cos θ / sin θ
  • sec θ = 1 / cos θ
  • csc θ = 1 / sin θ

Now, compute the left-hand side (LHS):

LHS = cot θ · sec θ = (cos θ / sin θ) · (1 / cos θ)

The cos θ terms cancel out:

LHS = (cos θ / sin θ) · (1 / cos θ) = 1 / sin θ

Recognize that 1 / sin θ is csc θ:

LHS = csc θ

Therefore, the identity holds:

cot θ · sec θ = csc θ

Paper For Above instruction

The task was to verify the trigonometric identity: cot θ · sec θ = csc θ. Using the fundamental definitions of cotangent, secant, and cosecant in terms of sine and cosine functions, the verification process involved substituting these definitions into the expression and simplifying. The calculations showed that cot θ · sec θ reduces to 1 / sin θ, which is precisely the definition of csc θ. This confirms that the identity is true for all θ where the functions are defined, i.e., where sin θ ≠ 0 and cos θ ≠ 0.

Understanding such identities is crucial in trigonometry because it allows the simplification of expressions and solutions to equations involving trigonometric functions. Verifying identities strengthens conceptual understanding of how these functions relate and interact, which is essential in fields like physics, engineering, and mathematics.

Overall, the proof reinforces that the relationships among basic trigonometric functions can be manipulated through substitution and algebraic simplification, providing valuable tools for solving more complex problems.

References

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  • Adams, R. A., & Essex, C. (2014). Calculus: A Complete Course. Pearson.
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  • Haggan, R., & Hauff, C. (2018). Mathematical Methods for Physics. Springer.
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