Trig identity
Prove:
cos2x tan(pie/4 - x)
------- =
1 + sin2x
To prove the given trigonometric identity:
cos2x / (1 + sin2x) = tan(π/4 - x)
We will begin by simplifying each side of the equation using trigonometric identities.
Starting with the left side:
cos2x / (1 + sin2x)
Now, we know the Pythagorean Identity, which states that:
sin2x + cos2x = 1
Rearranging this equation, we get:
cos2x = 1 - sin2x
Substituting this value into the original expression:
(1 - sin2x) / (1 + sin2x)
To simplify further, let's divide both the numerator and denominator by (1 + sin2x):
[(1 - sin2x) / (1 + sin2x)] / (1 + sin2x)
This simplifies to:
(1 - sin2x) / (1 + sin2x) * 1 / (1 + sin2x)
The denominator can also be written as:
(1 + sin2x) * (1 + sin2x)
Expanding this expression:
(1 + sin2x + sin2x + sin4x) / (1 + sin2x) * (1 + sin2x)
Canceling out the common factors:
(1 + sin4x) / (1 + sin2x)
Now, moving on to the right side of the equation:
tan(π/4 - x)
We know that the tangent function can be expressed as:
tan(a - b) = (tan(a) - tan(b)) / (1 + tan(a) * tan(b))
In our case, a = π/4 and b = x:
tan(π/4 - x) = (tan(π/4) - tan(x)) / (1 + tan(π/4) * tan(x))
The tangent of π/4 is 1, so we can substitute that:
tan(π/4 - x) = (1 - tan(x)) / (1 + tan(x))
Comparing this expression to the left side of the equation, we can see that they are identical:
(1 + sin4x) / (1 + sin2x) = (1 - tan(x)) / (1 + tan(x))
Thus, the trigonometric identity is proved.