verify the following identity used in calculus: cos(x+h)-cos(x)/h=cos(x)[cos(h)-1/h]-sin(x)[sin(h)/h]
To verify the given identity, we'll start by manipulating the left-hand side (LHS) of the equation and simplify it step by step until we reach the right-hand side (RHS) of the equation.
LHS: (cos(x+h) - cos(x)) / h
Step 1: Expand the cos(x+h) term using the angle sum formula.
LHS: ((cos(x)cos(h) - sin(x)sin(h)) - cos(x)) / h
Step 2: Simplify by combining like terms.
LHS: (cos(x)cos(h) - cos(x) - sin(x)sin(h)) / h
Step 3: Factor out cos(x) from the first two terms.
LHS: cos(x)(cos(h) - 1) - sin(x)sin(h) / h
Now let's work on the RHS of the equation.
RHS: cos(x)(cos(h) - 1) - sin(x)(sin(h) / h
Comparing the LHS and RHS:
LHS: cos(x)(cos(h) - 1) - sin(x)sin(h) / h
RHS: cos(x)(cos(h) - 1) - sin(x)(sin(h) / h
The LHS and RHS are identical. Therefore, the given identity is verified.
Note: In calculus, it's important to have a strong foundation in trigonometric identities. Being familiar with the angle sum and difference formulas, as well as other trigonometric identities, will help in simplifying expressions and verifying identities.