Factor the trigonometric expression.
[tex](\sin x-\csc x)^2-(\sin x+\csc x)^2 = [/tex] $\square$ (Simplify your answer.)
Answer (2)
Expand ( sin x − csc x ) 2 to sin 2 x − 2 sin x csc x + csc 2 x and ( sin x + csc x ) 2 to sin 2 x + 2 sin x csc x + csc 2 x .
Substitute the expanded forms into the original expression and distribute the negative sign.
Combine like terms, canceling out sin 2 x and csc 2 x .
Simplify using the identity csc x = s i n x 1 to get the final answer: − 4 .
Explanation
Expanding the Expression We are asked to factor and simplify the trigonometric expression ( sin x − csc x ) 2 − ( sin x + csc x ) 2 . Let's start by expanding the squared terms.
Expanding the Squares Expanding the first term, we have: ( sin x − csc x ) 2 = sin 2 x − 2 sin x csc x + csc 2 x Expanding the second term, we have: ( sin x + csc x ) 2 = sin 2 x + 2 sin x csc x + csc 2 x
Substituting and Distributing Now, substitute these expansions back into the original expression: ( sin 2 x − 2 sin x csc x + csc 2 x ) − ( sin 2 x + 2 sin x csc x + csc 2 x ) Distribute the negative sign: = sin 2 x − 2 sin x csc x + csc 2 x − sin 2 x − 2 sin x csc x − csc 2 x
Combining Like Terms Combine like terms. Notice that sin 2 x and csc 2 x cancel out: = − 2 sin x csc x − 2 sin x csc x = − 4 sin x csc x
Using Trigonometric Identity Recall that csc x = s i n x 1 . Substitute this into the expression: = − 4 sin x ⋅ sin x 1 = − 4
Final Answer Therefore, the simplified expression is − 4 .
Examples
Understanding trigonometric identities and factoring is crucial in fields like physics and engineering, especially when dealing with wave phenomena. For instance, simplifying complex expressions involving trigonometric functions can help in analyzing the behavior of light waves in optics or sound waves in acoustics. By factoring and simplifying, engineers can design more efficient systems for signal processing or image recognition, where trigonometric functions play a vital role in representing and manipulating data.
The trigonometric expression ( sin x − csc x ) 2 − ( sin x + csc x ) 2 simplifies to − 4 after expanding, substituting, and using trigonometric identities. The expansion leads to cancellations that allow for this simple result. Thus, the final result is − 4 .
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