Verify
Sinx-sinxcos^2x =sin^3x
Need every step!!!
To verify the identity sin(x) - sin(x)cos^2(x) = sin^3(x), we will work through the problem step by step:
Step 1: Start with the left-hand side of the equation: sin(x) - sin(x)cos^2(x).
Step 2: Factor out common terms. In this case, sin(x) is common to both terms, so we can factor it out:
sin(x) - sin(x)cos^2(x) = sin(x)(1 - cos^2(x)).
Step 3: Recall the trigonometric identity sin^2(x) + cos^2(x) = 1. Using this identity, we can rewrite 1 - cos^2(x) as sin^2(x):
sin(x)(1 - cos^2(x)) = sin(x)sin^2(x).
Step 4: Multiply the terms with the same base, sin(x):
sin(x)sin^2(x) = sin^3(x).
Step 5: Simplify the expression on the right-hand side to obtain sin^3(x).
Thus, we have verified that sin(x) - sin(x)cos^2(x) = sin^3(x).
To verify the given equation, we will simplify the left-hand side (LHS) and the right-hand side (RHS) of the equation separately and then equate them to check if they are equal.
Starting with the left-hand side (LHS):
Given: sin(x) - sin(x)cos^2(x)
Step 1: Rewrite cos^2(x) as (1 - sin^2(x))
LHS = sin(x) - sin(x)(1 - sin^2(x))
Step 2: Distribute sin(x) to the terms inside the parentheses
LHS = sin(x) - sin(x) + sin^3(x)
Step 3: Simplify the first two terms
LHS = 0 + sin^3(x)
LHS = sin^3(x)
Moving on to the right-hand side (RHS):
Given: sin^3(x)
RHS = sin^3(x)
Now, we equate the LHS and RHS:
LHS = RHS
sin^3(x) = sin^3(x)
Since both the LHS and RHS are equal, the given equation sin(x) - sin(x)cos^2(x) = sin^3(x) is verified.
LS = sinx(1 - cos^2 x)
= sinx(sin^2 x)
= sin^3 x
= RS