2sin(180°+x)sin(90°+x)/cost^4x-sin^4x
2sin(180°+x)sin(90°+x)/cos^4x-sin^4x
2(-sinx)(cosx)/(cos^2x-sin^2x)(cos^2x+sin^2x)
-sin2x/cos2x
-tan2x
To simplify the expression 2sin(180°+x)sin(90°+x)/cost^4x-sin^4x, we can use the trigonometric identities and properties to simplify each term individually:
Step 1: Simplify sin(180° + x):
Using the identity sin(180° + x) = -sin(x), the expression becomes -2sin(x).
Step 2: Simplify sin(90° + x):
Using the identity sin(90° + x) = cos(x), the expression becomes -2sin(x)cos(x).
Step 3: Expand the denominator:
Using the identity cos^2(x) = 1 - sin^2(x), we can rewrite the denominator as (1 - sin^2(x))^2 - sin^4(x).
Step 4: Simplify the denominator:
Expanding (1 - sin^2(x))^2 gives us (1 - 2sin^2(x) + sin^4(x)). Combining like terms with -sin^4(x), the denominator simplifies to 1 - 2sin^2(x) + sin^4(x) - sin^4(x). This further simplifies to 1 - 2sin^2(x).
Now, putting it all together, the simplified expression becomes:
(-2sin(x)cos(x)) / (1 - 2sin^2(x))
Therefore, the simplified expression is -2sin(x)cos(x) / (1 - 2sin^2(x)).
To simplify the given expression, we can use trigonometric identities and simplify each term step by step.
First, let's simplify the numerator: 2sin(180°+x)sin(90°+x).
Using the identity sin(a+b) = sin(a)cos(b) + cos(a)sin(b), we can rewrite sin(180°+x) as sin(180°)cos(x) + cos(180°)sin(x). Since sin(180°) = 0 and cos(180°) = -1, the term simplifies to -sin(x).
Next, we can rewrite sin(90°+x) using the same identity: sin(90°+x) = sin(90°)cos(x) + cos(90°)sin(x). Since sin(90°) = 1 and cos(90°) = 0, the term simplifies to cos(x).
Therefore, the numerator becomes -sin(x) cos(x).
Now, let's simplify the denominator: cost^4x - sin^4x.
Using the identity sin^2(x) = 1 - cos^2(x), we can rewrite sin^4x as (1 - cos^2x)^2. Expanding this equation, we get sin^4x = 1 - 2cos^2x + cos^4x.
Now, substitute this back into the original equation: cost^4x - sin^4x = cos^4x - (1 - 2cos^2x + cos^4x).
Simplifying this expression, we get cost^4x - sin^4x = -1 + 2cos^2x.
Therefore, the simplified expression is (-sin(x)cos(x))/(2cos^2x - 1).
So the final simplified expression is: (-sin(x)cos(x))/(2cos^2x - 1).