To prove the identity

MATHEMATICS · Single correct · CBSE · Mathematics · Introduction to Trigonometry · Introduction to Trigonometry

Problem

To prove the identity sin⁡θ1+cos⁡θ=1−cos⁡θsin⁡θ\dfrac{\sin\theta}{1+\cos\theta}=\dfrac{1-\cos\theta}{\sin\theta}, a student multiplies the numerator and denominator of the left-hand side by (1−cos⁡θ)(1-\cos\theta) and gets sin⁡θ(1−cos⁡θ)1−cos⁡2θ\dfrac{\sin\theta(1-\cos\theta)}{1-\cos^2\theta}. Which of the following next steps will lead directly to the right-hand side? (a) Factor the denominator as (1−cos⁡θ)(1+cos⁡θ)(1-\cos\theta)(1+\cos\theta) and cancel the common (1−cos⁡θ)(1-\cos\theta) term (b) Replace 1−cos⁡2θ1-\cos^{2}\theta with tan⁡2θ\tan^{2}\theta and cancel factors (c) Replace 1−cos⁡2θ1-\cos^{2}\theta with sin⁡2θ\sin^{2}\theta and cancel one sin⁡θ\sin\theta (d) Replace 1−cos⁡2θ1-\cos^{2}\theta with sin⁡2θ\sin^{2}\theta and rewrite the numerator as (1−cos⁡θ)2(1-\cos\theta)^{2}

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