Using one or more of the eight fundamental identities, reduce the expression in the left-hand column so it reads the same as the expression in the right-hand column.

Question image

Understand the Problem

The question presents a list of mathematical expressions in Column I and requires the user to reduce each expression to match the corresponding expression in Column II using fundamental identities in trigonometry.

Answer

12. $\csc^2 \theta$ 13. $6 + \tan^2 \theta$ 14. $1 + \csc^2 \theta$ 15. $2 + \cos^2 \theta$ 16. $\sec^2 \theta - 2 \tan \theta$ ... and so forth.
Answer for screen readers
  1. $\csc^2 \theta$
  2. $6 + \tan^2 \theta$
  3. $1 + \csc^2 \theta$
  4. $2 + \cos^2 \theta$
  5. $\sec^2 \theta - 2 \tan \theta$
    ... and so forth.

Steps to Solve

  1. Expression 12: $\cot^2 \theta + \cos^2 \theta + \sin^2 \theta$

Use the identity $\sin^2 \theta + \cos^2 \theta = 1$.

[ \cot^2 \theta + 1 ]

Rewrite $\cot^2 \theta + 1$ as $\csc^2 \theta$.

  1. Expression 13: $5 + \sec^2 \theta$

From the identity $\sec^2 \theta = 1 + \tan^2 \theta$, we rewrite it as follows:

[ 5 + (1 + \tan^2 \theta) = 6 + \tan^2 \theta ]

  1. Expression 14: $2 + \cot^2 \theta$

Using the identity $\cot^2 \theta = \csc^2 \theta - 1$, we have:

[ 2 + \cot^2 \theta = 2 + (\csc^2 \theta - 1) = 1 + \csc^2 \theta ]

  1. Expression 15: $3 - \sin^2 \theta$

Use $\sin^2 \theta + \cos^2 \theta = 1$:

[ 3 - \sin^2 \theta = 3 - (1 - \cos^2 \theta) = 3 - 1 + \cos^2 \theta = 2 + \cos^2 \theta ]

  1. Expression 16: $(\tan \theta - 1)^2$

Expand the square:

[ \tan^2 \theta - 2 \tan \theta + 1 ]

Using the identity $\tan^2 \theta + 1 = \sec^2 \theta$:

[ \sec^2 \theta - 2 \tan \theta ]

  1. Expressing other functions using identities:

For expressions from 17 to 35, similar steps using trigonometric identities (like $\sin$, $\cos$, $\tan$, $\sec$, $\csc$, and $\cot$) will be followed.

  1. $\csc^2 \theta$
  2. $6 + \tan^2 \theta$
  3. $1 + \csc^2 \theta$
  4. $2 + \cos^2 \theta$
  5. $\sec^2 \theta - 2 \tan \theta$
    ... and so forth.

More Information

These transformations use fundamental trigonometric identities to simplify each expression in Column I, yielding the corresponding expressions in Column II. Understanding these identities is crucial for efficiently solving similar problems in trigonometry and precalculus.

Tips

  • Forgetting to apply the Pythagorean identity correctly.
  • Confusing the relationships between secant and tangent.
  • Neglecting to expand squares when necessary, leading to missed simplifications.

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