sin(pi/2+x)cos(pi-x)cot(3pi/+x)=sin(pi/2-x)sin(3pi/2-x)cot(pi/2+x
use your sum/formulas. For example,
sin(pi/2+x) = cosx
cos(pi-x) = -cosx
and so on. Then the products will be much easier to manipulate.
To simplify the given expression, we can use the trigonometric identities to rewrite each term and then simplify the expression further.
1. Start with the left side of the equation: sin(pi/2 + x) * cos(pi - x) * cot(3pi/2 + x).
2. Use the sum and difference identities to replace the first term sin(pi/2 + x):
sin(pi/2 + x) = sin(pi/2) * cos(x) + cos(pi/2) * sin(x) = 1 * cos(x) + 0 * sin(x) = cos(x).
Therefore, the expression becomes:
cos(x) * cos(pi - x) * cot(3pi/2 + x).
3. Use the difference identity to replace the second term cos(pi - x):
cos(pi - x) = cos(pi) * cos(x) + sin(pi) * sin(x) = -1 * cos(x) + 0 * sin(x) = -cos(x).
The expression becomes:
cos(x) * (-cos(x)) * cot(3pi/2 + x).
4. Use the cotangent identity to replace cot(3pi/2 + x):
cot(3pi/2 + x) = cos(3pi/2 + x) / sin(3pi/2 + x).
Since cos(3pi/2 + x) = -cos(x) and sin(3pi/2 + x) = -sin(x), the expression becomes:
cos(x) * (-cos(x)) * (-cos(x) / -sin(x)).
Simplifying further:
cos(x) * (cos(x)) * (cos(x) / sin(x)).
5. Rearrange the expression for easier combination:
cos(x) * cos(x) * cos(x) / sin(x).
Now, let's simplify the right side of the equation: sin(pi/2 - x) * sin(3pi/2 - x) * cot(pi/2 + x).
1. Use the difference identity to replace the first term sin(pi/2 - x):
sin(pi/2 - x) = sin(pi/2) * cos(x) - cos(pi/2) * sin(x) = 1 * cos(x) + 0 * sin(x) = cos(x).
Therefore, the expression becomes:
cos(x) * sin(3pi/2 - x) * cot(pi/2 + x).
2. Use the difference identity to replace the second term sin(3pi/2 - x):
sin(3pi/2 - x) = sin(3pi/2) * cos(x) - cos(3pi/2) * sin(x) = -1 * cos(x) + 0 * sin(x) = -cos(x).
The expression becomes:
cos(x) * (-cos(x)) * cot(pi/2 + x).
3. Use the cotangent identity to replace cot(pi/2 + x):
cot(pi/2 + x) = cos(pi/2 + x) / sin(pi/2 + x).
Since cos(pi/2 + x) = 0 and sin(pi/2 + x) = 1, the expression becomes:
cos(x) * (-cos(x)) * (0 / 1) = 0.
Therefore, the left side of the equation simplifies to cos(x)^3 / sin(x), and the right side simplifies to 0.
So, the simplified equation is:
cos(x)^3 / sin(x) = 0.