WEBVTT

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Here is the problem: (1 minus cos(x))(1 plus cos(x)) equals sin(x) squared.

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Prove a trigonometric identity.

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Rewrite everything in terms of sin x and cos x. The expression already uses only sine and cosine.

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Left side in sine and cosine.

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Right side in sine and cosine.

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Subtract the sides and write the difference over a common denominator. The identity is true exactly when this numerator is zero for every value of the variable.

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Replace every cos squaredx with 1 minus sin squaredx and collect terms.

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The numerator is identically zero. Every term cancelled after using sin squared plus cos squared equals 1, so both sides are equal wherever they are defined.

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So the answer is The identity is true (for every value where both sides are defined).

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After rewriting both sides with sine and cosine and replacing cos squared with 1 minus sin squared, the difference of the two sides is exactly 0, so (1 minus cos(x))(1 plus cos(x)) equals sin(x) squared holds for every x at which both sides are defined.

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Numeric spot check at 10 sample values: both sides agree.