Problem 49 - Precalculus Essentials

Question

Verify each identity. $$\frac{\cos (x+h)-\cos x}{h}=\cos x \frac{\cos h-1}{h}-\sin x \frac{\sin h}{h}$$

Step-by-Step Solution

Answer

Yes, by applying trigonometric product-to-sum identities, factoring and dividing by 'h', the right hand side of the equation could be perfectly matched with the left hand side, thus verifying the identity.

Step 1: Expand the RHS

Expand the right hand side (RHS) of the equation using trigonometric product-to-sum identities: \( \cos a \cos b - \sin a \sin b = \cos (a + b) \). So RHS = \( \cos x \cos h - \cos x - \sin x \sin h \) .

Step 2: Factor out \( \cos x \)

Next, factor out \( \cos x \) from the first two terms on the RHS to get: \( \cos x (\cos h - 1) - \sin x \sin h \).

Step 3: Compare the RHS with LHS

Finally, divide both of these parts by 'h', producing \( \cos x \frac{\cos h -1}{h} - \sin x \frac{\sin h}{h} \) , which exactly matches the left hand side (LHS) of the given equation, hence the identity is verified.