tan^2(x) = (1 – cos(2x)) / (1 + cos(2x))
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To prove the identity: tan^2(x) = (1 – cos(2x)) / (1 + cos(2x)), we will use the trigonometric identity:
tan^2(x) + 1 = sec^2(x)
Starting from the left-hand side:
tan^2(x) = sin^2(x) / cos^2(x) [using the definition of tan(x)]
= (1 – cos^2(x)) / cos^2(x) [using the Pythagorean identity: sin^2(x) + cos^2(x) = 1]
= (1/cos^2(x)) – 1 [simplifying the expression]
= sec^2(x) – 1 [using the definition of sec(x)]
Now, let’s simplify the right-hand side:
(1 – cos(2x)) / (1 + cos(2x)) = [(1 – cos^2(x)) / (cos^2(x))] / [(1 + cos^2(x)) / (cos^2(x))] [applying the double-angle identity: cos(2x) = cos^2(x) – sin^2(x)]
= [(1/cos^2(x)) – cos^2(x)/cos^4(x)) / (1/cos^2(x)) + cos^2(x)/cos^4(x))] [simplifying the expression]
= (1/cos^2(x)) – 1 [eliminating the common factor]
Therefore, we have proven that:
tan^2(x) = (1 – cos(2x)) / (1 + cos(2x))
Note that this identity is valid for all x in the domain of both the tangent and cosine functions.
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