Prove that, wherever both sides are defined: (1 + tan² x) cos² x = 1; (sin x)/(1 + cos x) + (1 +…

Step 1. Part (a). $(1 + \tan^2 x)\cos^2 x = \sec^2 x \cos^2 x = \dfrac{1}{\cos^2 x} \times \cos^2 x = 1$.

Step 2. Part (b), the numerator. Over the common denominator, the numerator is $\sin^2 x + (1 + \cos x)^2 = \sin^2 x + 1 + 2\cos x + \cos^2 x$.

Step 3. Use the identity. Since $\sin^2 x + \cos^2 x = 1$, the numerator is $2 + 2\cos x = 2(1 + \cos x)$.

Step 4. Cancel. $\dfrac{2(1 + \cos x)}{\sin x(1 + \cos x)} = \dfrac{2}{\sin x} = 2\operatorname{cosec} x$.