Questions: (csc(x))^2-(cot(x))^2=

(csc(x))^2-(cot(x))^2=
Transcript text: \[ (\csc (x))^{2}-(\cot (x))^{2}= \]
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Solution

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Solution Steps

Solution Approach

To solve the given trigonometric identity, we can use the Pythagorean identity for cosecant and cotangent. Specifically, we know that: \[ (\csc(x))^2 = 1 + (\cot(x))^2 \] Using this identity, we can simplify the given expression.

Step 1: Define the Expression

We start with the expression given in the problem: \[ (\csc(x))^2 - (\cot(x))^2 \]

Step 2: Apply the Pythagorean Identity

Using the identity \( (\csc(x))^2 = 1 + (\cot(x))^2 \), we can rewrite the expression: \[ (\csc(x))^2 - (\cot(x))^2 = (1 + (\cot(x))^2) - (\cot(x))^2 \]

Step 3: Simplify the Expression

Upon simplifying, we find: \[ 1 + (\cot(x))^2 - (\cot(x))^2 = 1 \]

Final Answer

Thus, the simplified expression is: \[ \boxed{1} \]

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