show that :
cos(360*+x)-tan(150*-x)sin(360*-2x)cos(-x)/ sin(90*+x)=cos2x
To prove that cos(360*+x) - tan(150*-x)sin(360*-2x)cos(-x)/ sin(90*+x) = cos(2x), we need to simplify both sides of the equation and show that they are equal.
Let's start by simplifying the left side of the equation step by step:
cos(360*+x) - tan(150*-x)sin(360*-2x)cos(-x)/ sin(90*+x)
1. Simplify the expression cos(360*+x):
Since cos(360*) is equal to 1, we can simplify cos(360*+x) to cos(x):
cos(x) - tan(150*-x)sin(360*-2x)cos(-x)/ sin(90*+x)
2. Simplify the expression sin(360*-2x):
Since sin(360* * n) is equal to 0 for any integer value of n, we can simplify sin(360*-2x) to 0:
cos(x) - tan(150*-x)(0)cos(-x)/ sin(90*+x)
3. Simplify the expression tan(150*-x):
We know that tan(180*+θ) is equal to tan(θ), so we can write tan(150*-x) as tan(-x):
cos(x) - tan(-x)(0)cos(-x)/ sin(90*+x)
4. Simplify the expression tan(-x):
Since tan(-x) is equal to -tan(x), we can rewrite tan(-x) as -tan(x):
cos(x) - (-tan(x))(0)cos(-x)/ sin(90*+x)
5. Simplify the expression (0)cos(-x):
Any value multiplied by 0 is equal to 0, so we can simplify (0)cos(-x) to 0:
cos(x) - (-tan(x))(0)/ sin(90*+x)
6. Simplify the expression (-tan(x))(0):
Any value multiplied by 0 is equal to 0, so we can simplify (-tan(x))(0) to 0:
cos(x) - 0/ sin(90*+x)
7. Simplify the expression 0/ sin(90*+x):
Any value divided by 0 is undefined, so 0/ sin(90*+x) is undefined:
cos(x) - undefined
Now let's simplify the right side of the equation:
cos(2x)
Since there are no additional terms or operations to perform, we can leave cos(2x) as it is.
Therefore, the left side of the equation is cos(x) - undefined, while the right side of the equation is cos(2x). Since the left side is undefined and the right side is equal to cos(2x), we can conclude that the equation is not true.
Hence, the statement "cos(360*+x) - tan(150*-x)sin(360*-2x)cos(-x)/ sin(90*+x) = cos(2x)" is not valid.