Question

Prove the identity. csc⁴ t-cot⁴ t=csc² t+cot² t

Answer 1

To prove the trigonometric identity csc⁴ t - cot⁴ t = csc² t + cot² t, we rewrote each term in terms of sine and cosine, factored the difference of squares, and showed that both sides of the identity are equal.

Step-by-step explanation:

The question asks to prove the trigonometric identity csc⁴ t - cot⁴ t = csc² t + cot² t. We start by writing all terms in terms of sine and cosine to facilitate simplification. Recognize that csc t is the reciprocal of sin t, and cot t is the reciprocal of tan t, which is also cos t divided by sin t.

First, express the csc and cot functions in terms of sine and cosine:

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• csc t = 1/sin t
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• cot t = cos t/sin t

Then square both sides of each equation:

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• csc² t = 1/sin² t
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• cot² t = cos² t/sin² t

Now, we expand the left side of the given identity:

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• csc⁴ t - cot⁴ t = (csc² t)² - (cot² t)²
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• = (1/sin² t)² - (cos² t/sin² t)²
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• = 1/sin⁴ t - cos⁴ t/sin⁴ t
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• = (1 - cos⁴ t)/sin⁴ t

Now factor the numerator using the difference of squares:

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• 1 - cos⁴ t = (1 + cos² t)(1 - cos² t)
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• = (1 + cos² t)(sin² t), since 1 - cos² t = sin² t

Substitute this back into the identity:

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• (1 - cos⁴ t)/sin⁴ t = (1 + cos² t)(sin² t)/sin⁴ t
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• = (1 + cos² t)/sin² t
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• = 1/sin² t + cos² t/sin² t
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• = csc² t + cot² t

Hence, we have shown that csc⁴ t - cot⁴ t indeed equals csc² t + cot² t, proving the identity.