Verify the identity:
sin^(1/2)x*cosx - sin^(5/2)*cosx = cos^3x sq root sin x
I honestly have no clue how to approach the sin^(5/2)*cosx part of the equation
since 5/2 = 2 + 1/2, you have
u^5/2 = u^2 * u^1/2, and so,
√sinx cosx - sin^2x √sinx cosx
√sinx cosx (1-sin^2 x)
√sinx cosx cos^2x
√sinx cos^3x
To verify the identity, let's break down the expression and simplify it step by step.
Expression: sin^(1/2)x * cosx - sin^(5/2)x * cosx
Step 1: Recall the exponent rules
The exponent rule states that (a^m)^n = a^(m*n). Therefore, we can rewrite sin^(1/2)x as (sinx)^(1/2) and sin^(5/2)x as (sinx)^(5/2).
Expression: (sinx)^(1/2) * cosx - (sinx)^(5/2) * cosx
Step 2: Factor out common terms
Both terms have cosx, so we can factor it out:
Expression: cosx * [(sinx)^(1/2) - (sinx)^(5/2)]
Step 3: Simplify the exponent in each term
To simplify the exponent, we can use the power rule for exponents, which states that (a^m)^n = a^(m*n). Applying this rule, we have:
Expression: cosx * [sin^(1/4)x - sin^(5/4)x]
Step 4: Rewrite the radicals as fractional exponents
Let's express the square roots as fractional exponents: sin^(1/4)x = (sinx)^(1/4) and sin^(5/4)x = (sinx)^(5/4).
Expression: cosx * [(sinx)^(1/4) - (sinx)^(5/4)]
Step 5: Factor out the common term
Both terms have (sinx)^(1/4), so we can factor it out:
Expression: (sinx)^(1/4) * cosx * [1 - (sinx)^(4/4)]
Step 6: Simplify the exponent in the second term
Since the exponent is 4/4, which is equal to 1, we can simplify it to (sinx)^1 = sinx.
Expression: (sinx)^(1/4) * cosx * [1 - sinx]
Step 7: Simplify the expression inside the square brackets
Distribute the multiplication:
Expression: (sinx)^(1/4) * cosx - (sinx)^(1/4) * cosx * sinx
Step 8: Rearrange the terms
The two terms have a common factor of (sinx)^(1/4) * cosx, so we can factor it out:
Expression: (sinx)^(1/4) * cosx * (1 - sinx)
Step 9: Simplify the expression
The expression (1 - sinx) is equal to cos^2x using the identity sin^2x + cos^2x = 1.
Expression: (sinx)^(1/4) * cosx * cos^2x
Step 10: Simplify the final expression
Using the property of exponents, (a^m) * a^n = a^(m + n), we can simplify the expression (sinx)^(1/4) * cosx * cos^2x:
Expression: (sinx)^(1/4) * cos^3x
Therefore, the given expression sin^(1/2)x * cosx - sin^(5/2)x * cosx simplifies to cos^3x * (sinx)^(1/4).
We have now verified the given identity: sin^(1/2)x * cosx - sin^(5/2)x * cosx = cos^3x * (sinx)^(1/4).