1. Prove (a) $\cos x \tan^3 x = \sin x \tan^2 x$. Step 1: Recall that $\tan x = \frac{\sin x}{\cos x}$. Step 2: Substitute $\tan x$ into the left side: $$\cos x \left(\frac{\sin x}{\cos x}\right)^3 = \cos x \frac{\sin^3 x}{\cos^3 x} = \frac{\cos x \sin^3 x}{\cos^3 x} = \frac{\sin^3 x}{\cos^2 x}$$ Step 3: Simplify the right side: $$\sin x \left(\frac{\sin x}{\cos x}\right)^2 = \sin x \frac{\sin^2 x}{\cos^2 x} = \frac{\sin^3 x}{\cos^2 x}$$ Step 4: Both sides equal $\frac{\sin^3 x}{\cos^2 x}$, so the identity holds. 2. Prove (b) $\sin^2 \theta + \cos^4 \theta = \cos^2 \theta + \sin^4 \theta$. Step 1: Rearrange the equation: $$\sin^2 \theta - \sin^4 \theta = \cos^2 \theta - \cos^4 \theta$$ Step 2: Factor both sides: $$\sin^2 \theta (1 - \sin^2 \theta) = \cos^2 \theta (1 - \cos^2 \theta)$$ Step 3: Use Pythagorean identity $1 - \sin^2 \theta = \cos^2 \theta$ and $1 - \cos^2 \theta = \sin^2 \theta$: $$\sin^2 \theta \cos^2 \theta = \cos^2 \theta \sin^2 \theta$$ Step 4: Both sides are equal, so the identity is true. 3. Prove (c) $ (\sin x + \cos x) \left(\frac{\tan^2 x + 1}{\tan x}\right) = \frac{1}{\cos x} + \frac{1}{\sin x}$. Step 1: Recall $\tan^2 x + 1 = \sec^2 x = \frac{1}{\cos^2 x}$. Step 2: Substitute into the left side: $$ (\sin x + \cos x) \frac{\frac{1}{\cos^2 x}}{\tan x} = (\sin x + \cos x) \frac{1}{\cos^2 x \tan x}$$ Step 3: Recall $\tan x = \frac{\sin x}{\cos x}$, so: $$\frac{1}{\cos^2 x \tan x} = \frac{1}{\cos^2 x \frac{\sin x}{\cos x}} = \frac{1}{\cos x \sin x}$$ Step 4: So left side becomes: $$ (\sin x + \cos x) \frac{1}{\cos x \sin x} = \frac{\sin x + \cos x}{\sin x \cos x}$$ Step 5: The right side is: $$ \frac{1}{\cos x} + \frac{1}{\sin x} = \frac{\sin x}{\sin x \cos x} + \frac{\cos x}{\sin x \cos x} = \frac{\sin x + \cos x}{\sin x \cos x}$$ Step 6: Both sides are equal, so the identity holds. 4. Prove (d) $\tan^2 \beta + \cos^2 \beta + \sin^2 \beta = \frac{1}{\cos^2 \beta}$. Step 1: Recall $\tan^2 \beta = \frac{\sin^2 \beta}{\cos^2 \beta}$ and $\sin^2 \beta + \cos^2 \beta = 1$. Step 2: Substitute into the left side: $$ \frac{\sin^2 \beta}{\cos^2 \beta} + 1 = \frac{\sin^2 \beta + \cos^2 \beta}{\cos^2 \beta} = \frac{1}{\cos^2 \beta}$$ Step 3: Left side equals right side, so the identity is true. Final answers: All four identities are proven true.