$\"\"$

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The trigonometric identity $\"\"$ .

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The sum-to-product formula: $\"\"$ .

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Consider $\"\"$ .

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Here $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$ .

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Consider $\"\"$ .

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The double-angle formula: $\"\"$ .

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$\"\"$ .

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$\"\"$

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Consider the left hand side identity $\"\"$ .

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Substitute $\"\"$ and $\"\"$ .

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$\"\"$

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The sum-to-product formula: $\"\"$ .

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$\"\"$

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= Right hand side of the identity.

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$\"\"$ .

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$\"\"$

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$\"\"$ .