To determine if the equation tan(2x) sec(2x) = 1 is an identity, we need to analyze both sides of the equation.
First, let’s recall the definitions of the tangent and secant functions:
- tan(θ) = sin(θ) / cos(θ)
- sec(θ) = 1 / cos(θ)
Applying these definitions to our equation:
tan(2x) = sin(2x) / cos(2x> and sec(2x) = 1 / cos(2x)
Substituting these into the left side of the equation leads to:
tan(2x) sec(2x) = (sin(2x) / cos(2x)) * (1 / cos(2x)) = sin(2x) / (cos(2x) * cos(2x))
Now, simplifying this further:
The expression simplifies to:
sin(2x) / cos²(2x)
For the right side of the equation, we have simply 1.
Now, we check if sin(2x) / cos²(2x) = 1.
This equation can be transformed into:
sin(2x) = cos²(2x)
This equality holds true only for specific values of x, rather than being true for all values, which means that tan(2x) sec(2x) = 1 is not an identity.
In conclusion, tan(2x) sec(2x) = 1 is not an identity as it does not hold true for all values of x.