Substitution techniques are often used while simplifying trigonometric expressions…

MATHEMATICS · Multi-part · CBSE · Mathematics · Introduction to Trigonometry · Introduction to Trigonometry

Problem

Substitution techniques are often used while simplifying trigonometric expressions containing powers of sine and cosine. (i) State the fundamental relationship that connects sin⁡2θ\sin^{2}\theta and cos⁡2θ\cos^{2}\theta. (ii) Explain, in words, why replacing sin⁡2θ\sin^{2}\theta with 1−cos⁡2θ1-\cos^{2}\theta (or vice-versa) can make an algebraic simplification easier when an expression involves both sin⁡2θ\sin^{2}\theta and cos⁡θ\cos\theta. (iii) Illustrate your explanation by taking the expression Q=sin⁡2θ+sin⁡θcos⁡θQ = \sin^{2}\theta + \sin\theta\cos\theta and briefly outlining how such a substitution leads to a simpler form of QQ.

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