Consider the four standard products that involve and .

MATHEMATICS · Multi-part · CBSE · Mathematics · Trigonometric Functions · Trigonometric Functions

Problem

Consider the four standard products that involve sin⁡x, cos⁡x, sin⁡y\sin x,\,\cos x,\,\sin y and cos⁡y\cos y. (i) Write sin⁡x cos⁡y+cos⁡x sin⁡y\sin x\,\cos y+\cos x\,\sin y as a single trigonometric function of a single angle. (ii) Write cos⁡x cos⁡y−sin⁡x sin⁡y\cos x\,\cos y-\sin x\,\sin y as a single trigonometric function of a single angle. (iii) Give a brief geometrical or algebraic explanation (2–3 sentences) to show why the sign pattern in the product determines whether the compressed angle is (x+y)(x+y) or (x−y)(x-y). (iv) Without further derivation, state the single-angle equivalents of sin⁡x cos⁡y−cos⁡x sin⁡y\sin x\,\cos y-\cos x\,\sin y and of cos⁡x cos⁡y+sin⁡x sin⁡y\cos x\,\cos y+\sin x\,\sin y. Explain in one sentence how your answer in part (iii) helps you decide the signs.

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