Question
Establish the identity. \[ (\csc \theta+\cot \theta)(\csc \theta-\cot \theta)=1 \] Multiply and write the left side expression as the difference of two squares.
Ask by Smith Smith. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
(\csc \theta + \cot \theta)(\csc \theta - \cot \theta) = 1
\]
Solution
To establish the trigonometric identity:
\[
(\csc \theta + \cot \theta)(\csc \theta - \cot \theta) = 1
\]
we can utilize the **difference of squares** formula, which states that:
\[
(a + b)(a - b) = a^2 - b^2
\]
Let's apply this formula to the left side of the identity:
\[
(\csc \theta + \cot \theta)(\csc \theta - \cot \theta) = \csc^2 \theta - \cot^2 \theta
\]
Now, recall the fundamental Pythagorean identity in trigonometry:
\[
\csc^2 \theta = 1 + \cot^2 \theta
\]
Using this identity, we can substitute \( \csc^2 \theta \):
\[
\csc^2 \theta - \cot^2 \theta = (1 + \cot^2 \theta) - \cot^2 \theta = 1
\]
**Conclusion:**
\[
(\csc \theta + \cot \theta)(\csc \theta - \cot \theta) = \csc^2 \theta - \cot^2 \theta = 1
\]
This confirms that the original identity holds true.
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To establish the identity, let's first rewrite the left-hand side as the difference of two squares. We can express the left side as: \[ (\csc \theta + \cot \theta)(\csc \theta - \cot \theta) = \csc^2 \theta - \cot^2 \theta \] Now, we can utilize the Pythagorean identity: \[ \csc^2 \theta = 1 + \cot^2 \theta \] Using this identity, we substitute \(\csc^2 \theta\) into our expression: \[ \csc^2 \theta - \cot^2 \theta = (1 + \cot^2 \theta) - \cot^2 \theta = 1 \] Thus, we have shown that: \[ (\csc \theta + \cot \theta)(\csc \theta - \cot \theta) = 1 \] This confirms the identity is true!
