Problem 50
Question
Verify each identity. $$\frac{\sin (x+h)-\sin x}{h}=\cos x \frac{\sin h}{h}+\sin x \frac{\cos h-1}{h}$$
Step-by-Step Solution
Verified Answer
Hence, we have verified the given identity.
1Step 1: Expand the numerator
Expand the expression \( \sin(x+h) \) in the numerator using the sum of angles formula for sine, to get \( \sin(x)\cos(h) + \cos(x)\sin(h) \). Then we subtract \( \sin(x) \) from it.
2Step 2: Rearrange Terms
Rearrange the expression obtained from Step 1 as \( [\sin(x)\cos(h) - \sin(x)] + \cos(x)\sin(h) \).
3Step 3: Factor out common terms
In the step 2 rearrangement, we can factor out the common terms to get \( \sin(x)[\cos(h) - 1] + \cos(x)\sin(h) \).
4Step 4: Apply limits
We know that the limit as \( h \) approaches 0 of \( \frac{\sin(h)}{h} \) is 1 and the limit as \( h \) approaches 0 of \( \frac{\cos(h)-1}{h} \) is 0. Therefore applying these limits and simplifying, we obtain the identity we needed to prove.
Other exercises in this chapter
Problem 50
Verify each identity. $$\frac{\cos ^{2} t+4 \cos t+4}{\cos t+2}=\frac{2 \sec t+1}{\sec t}$$
View solution Problem 50
Involve trigonometric equations quadratic in form. Solve each equation on the interval \([0,2 \pi)\) $$3 \tan ^{2} x-9=0$$
View solution Problem 51
In this section, we saw how sums could be expressed as products. Sums of trigonometric functions can also be used to describe functions that are not trigonometr
View solution Problem 51
Verify each identity. $$\cos ^{4} t-\sin ^{4} t=1-2 \sin ^{2} t$$
View solution