Verify tanxsin2x = 2-2cos^2x
tan(x) = sin(x) / cos(x)
sin(2x) = 2 * sin(x) * cos(x)
tan(x)*sin(2x) = [sin(x) / cos(x)] * 2 * sin(x) * cos(x) =
2 sin(x) * sin(x) = 2 sin^2(x)
sin^2(x) + cos^2(x) = 1
sin^(x) = 1 - cos^2(x)
2 sin^2(x) = 2 * [ 1 - cos^2(x) ] = 2 - 2 * cos^2(x)
Sure, let's verify this equation in a fun and clownish way!
First, let's start with the left side of the equation:
tan(x) * sin(2x)
Well, tan(x) can be expressed as sin(x) / cos(x), so we can rewrite the equation as:
sin(x) / cos(x) * sin(2x)
Now, let's use the double angle identity for sin(2x):
sin(2x) = 2sin(x)cos(x)
So, substituting the double angle identity into our equation, we get:
sin(x) / cos(x) * 2sin(x)cos(x)
Now, let's simplify this expression a bit:
2sin(x) * sin(x) * cos(x) / cos(x)
Notice that the cos(x) in the numerator and denominator cancel out, leaving us with:
2sin^2(x)
Now, let's compare it to the right side of the equation:
2 - 2cos^2(x)
We can rewrite it as:
2(1 - cos^2(x))
Applying the identity sin^2(x) + cos^2(x) = 1, we get:
2sin^2(x)
Great! We've reached the same expression as on the left side of the equation. So, it is confirmed that tan(x) * sin(2x) = 2 - 2cos^2(x).
Guess math and clowns can have their fun together!
To verify the trigonometric identity tan(x)sin^2(x) = 2 - 2cos^2(x), we will manipulate the left-hand side of the equation and simplify it to match the right-hand side.
Starting with the left-hand side of the equation:
tan(x)sin^2(x)
We know that tan(x) can be expressed as sin(x)/cos(x), so we substitute that in:
(sin(x)/cos(x)) * sin^2(x)
Next, we simplify by multiplying the fractions:
(sin(x) * sin^2(x)) / cos(x)
Next, we simplify the numerator by using the trigonometric identity sin^2(x) = 1 - cos^2(x):
(sin(x) * (1 - cos^2(x))) / cos(x)
Now we distribute sin(x) to the terms inside the parentheses:
(sin(x) - sin(x) * cos^2(x)) / cos(x)
Expanding further:
sin(x)/cos(x) - sin(x) * cos^2(x)/cos(x)
Using the identity sin(x)/cos(x) = tan(x):
tan(x) - tan(x) * cos^2(x)
Factoring out tan(x):
tan(x) * (1 - cos^2(x))
Notice that 1 - cos^2(x) is equivalent to sin^2(x):
tan(x) * sin^2(x)
Comparing this result to the right-hand side of the identity, we can see that they match:
tan(x) * sin^2(x) = tan(x) * sin^2(x) = 2 - 2cos^2(x)
Therefore, we have successfully verified the trigonometric identity.
To verify the given equation: tan(x)sin(2x) = 2 - 2cos^2(x), we will simplify the left-hand side (LHS) and the right-hand side (RHS) separately and then see if they are equal.
Let's start with the LHS:
LHS = tan(x)sin(2x)
The trigonometric identity we will use here is sin(2x) = 2sin(x)cos(x):
LHS = tan(x) * 2sin(x)cos(x)
Next, we'll rewrite tan(x) as sin(x)/cos(x):
LHS = (sin(x)/cos(x)) * 2sin(x)cos(x)
Now, there are two terms in the numerator and denominator of LHS that can be canceled out:
LHS = (sin(x) * 2sin(x))
Using the identity sin^2(x) = (1 - cos^2(x)), we can rewrite the above equation:
LHS = 2sin^2(x)
Moving on to the RHS:
RHS = 2 - 2cos^2(x)
Now that we have simplified both sides, let's check if they are equal:
LHS = 2sin^2(x)
RHS = 2 - 2cos^2(x)
By using the identity cos^ 2(x) = 1 - sin^ 2(x), we can rewrite the RHS:
RHS = 2 - 2(1 - sin^2(x))
RHS = 2 - 2 + 2sin^2(x)
RHS = 2sin^2(x)
We can see that the LHS and RHS are equal, which verifies the given equation:
LHS = RHS
2sin^2(x) = 2sin^2(x)
Hence, the equation tan(x)sin(2x) = 2 - 2cos^2(x) is verified.