Sin(x+y+z)+sin(x-y-z)+sin(x+y-z)+sin(x-y+z)Is equal to how much. Find the answer using transformations comcept

To solve the expression sin(x+y+z) + sin(x-y-z) + sin(x+y-z) + sin(x-y+z), we can utilize sine transformation properties. This approach will help us simplify the expression step by step.

Applying Sine Transformation

The sine function has a property that can be quite useful here. Specifically, we can use the fact that sin(a + b) = sin(a)cos(b) + cos(a)sin(b). However, in this case, we can also benefit from the sine addition and subtraction identities.

We can rearrange and group the terms in the expression:

• sin(x+y+z)
• sin(x+y-z)
• sin(x-y-z)
• sin(x-y+z)

Now, let’s pair the terms:

• sin(x+y+z) + sin(x-y+z)
• sin(x+y-z) + sin(x-y-z)

First Pairing: sin(x+y+z) + sin(x-y+z)

Using the sine addition formula, we can simplify this pair. The formula for the sum of two sine functions is:

sin(A) + sin(B) = 2sin((A+B)/2)cos((A-B)/2)

Let A = x+y+z and B = x-y+z:

• A + B = (x+y+z) + (x-y+z) = 2x + 2z
• A - B = (x+y+z) - (x-y+z) = 2y

Thus, we have:

sin(x+y+z) + sin(x-y+z) = 2sin((2x + 2z)/2)cos(2y/2)

This simplifies to:

2sin(x + z)cos(y)

Second Pairing: sin(x+y-z) + sin(x-y-z)

Now, for the second pair:

Let A = x+y-z and B = x-y-z:

• A + B = (x+y-z) + (x-y-z) = 2x - 2z
• A - B = (x+y-z) - (x-y-z) = 2y

Using the same sine addition formula, we get:

sin(x+y-z) + sin(x-y-z) = 2sin((2x - 2z)/2)cos(2y/2)

This simplifies to:

2sin(x - z)cos(y)

Combining Both Results

Now, we can combine the results from both pairs:

2sin(x + z)cos(y) + 2sin(x - z)cos(y)

Factoring out the common term, we get:

2cos(y)(sin(x + z) + sin(x - z))

Final Simplification

Applying the sine addition formula again to sin(x + z) + sin(x - z), we have:

sin(x + z) + sin(x - z) = 2sin(x)cos(z)

Thus, we can simplify the expression to:

2cos(y)(2sin(x)cos(z)) = 4sin(x)cos(z)cos(y)

The Result

The final simplified form of the original expression is:

4sin(x)cos(y)cos(z)

This result shows how we can use transformations and trigonometric identities to simplify complex expressions effectively.