SIMPLIFY
1. ((x/4)-(1/x)) / ((1/2x)+(1/4))
VERIFY
1. secXsin2X = (2-2cos^2X)/sinX
((x/4)-(1/x)) / ((1/2x)+(1/4))
((x^2-4)/4x) / ((2+x)/4x)
(x^2-4) / (x+2)
(x+2)(x-2)/(x+2)
x-2
secX*2sinXcosX = 2(1-cos^2X)/sinX
2sinX = 2sin^2X/sinX
2sinX = 2sinX
OMG thank you so much!
To simplify the expression ((x/4)-(1/x)) / ((1/2x)+(1/4)), we can follow these steps:
Step 1: Simplify the numerator and denominator separately.
In the numerator, we have (x/4)-(1/x). To simplify this, we find a common denominator, which is 4x. So, the expression becomes (x^2 - 4)/(4x).
In the denominator, we have (1/2x) + (1/4). To simplify this, we find a common denominator, which is 4x. So, the expression becomes (2 + x)/(4x).
Step 2: Divide the numerator by the denominator.
We can rewrite the expression as (x^2 - 4)/(4x) ÷ (2 + x)/(4x).
Step 3: Flip the denominator and multiply.
When dividing fractions, we can flip the second fraction and multiply. So, the expression becomes (x^2 - 4)/(4x) * (4x)/(2 + x).
Step 4: Simplify and cancel common factors.
The 4x in the numerator and denominator cancels out, leaving us with (x^2 - 4)/(2 + x).
Therefore, the simplified expression is (x^2 - 4)/(2 + x).
To verify the trigonometric identity sec(X)*sin(2X) = (2 - 2cos^2(X))/sin(X), we can follow these steps:
Step 1: Recall the trigonometric identities:
- sec(X) = 1/cos(X)
- sin(2X) = 2sin(X)cos(X)
- cos^2(X) = 1 - sin^2(X)
Step 2: Simplify the left-hand side (LHS) of the identity.
Using the identities mentioned above, we have:
sec(X)*sin(2X) = (1/cos(X)) * (2sin(X)cos(X))
= (2sin(X)cos(X))/(cos(X))
= 2sin(X)
Step 3: Simplify the right-hand side (RHS) of the identity.
Using the identity cos^2(X) = 1 - sin^2(X), we have:
(2 - 2cos^2(X))/sin(X) = (2 - 2(1 - sin^2(X)))/sin(X)
= (2 - 2 + 2sin^2(X))/sin(X)
= (2sin^2(X))/sin(X)
= 2sin(X)
Step 4: Compare LHS and RHS.
We have shown that LHS = 2sin(X) and RHS = 2sin(X), so the identity is verified.
Therefore, the identity sec(X)*sin(2X) = (2 - 2cos^2(X))/sin(X) is true.