sin2x − cos2x − tan2x = 2sin2x − 2sin4x − 1/
1 − sin2x
for x = 45
I can't quite make out your equation, but it is easy to plug in x=45
sin2x = sin90 = 1
sin4x = sin180 = 0
and so on.
If that still stumps you, show us how far you got, and try using some parentheses to make things clear, such as
1/(2-sin2x)
and say sin^2x or sin^2(x) if you mean sin squared of x instead of sin(2x)
To evaluate the expression for x = 45 degrees, we need to substitute the value of x into the given expression and simplify it step by step.
Let's start with the left-hand side (LHS):
sin2x - cos2x - tan2x
For x = 45 degrees, we have:
sin(2 * 45) - cos(2 * 45) - tan(2 * 45)
To evaluate the trigonometric functions for 45 degrees, we can use the special angles of the unit circle. For 45 degrees, we know that sin(45) = cos(45) = 1/sqrt(2) and tan(45) = 1.
Substituting these values into the expression, we get:
(1/sqrt(2))^2 - (1/sqrt(2))^2 - 1
Simplifying further:
1/2 - 1/2 - 1
Combining like terms:
-1
So, the LHS evaluates to -1.
Now, let's move to the right-hand side (RHS):
2sin2x - 2sin4x - 1 / 1 - sin2x
Again, substituting x = 45 degrees:
2sin(2 * 45) - 2sin(4 * 45) - 1 / 1 - sin(2 * 45)
Using the values of sin(45) and sin(90), we can simplify this further:
2(1/sqrt(2))^2 - 2(1/2)^2 - 1 / 1 - (1/sqrt(2))^2
Simplifying:
2/2 - 2/4 - 1 / 1 - 1/2
Combining like terms:
1 - 1/2 - 1 / 1/2
To subtract fractions, we need to have a common denominator:
1 - 1/2 - 2/2 / 1/2
Combining the numerators:
(2 - 2 - 2) / 2
Simplifying:
-2 / 2
Reducing the fraction:
-1
So, the RHS also evaluates to -1.
Therefore, the given equation is true for x = 45 degrees.