(t) [tex]y=\left(\frac{\sin 2 x}{1+\cos 2 x}\right)^2[/tex]
Answer (1)
Apply the double angle identities sin 2 x = 2 sin x cos x and cos 2 x = 2 cos 2 x − 1 .
Substitute into the function: y = ( 1 + ( 2 c o s 2 x − 1 ) 2 s i n x c o s x ) 2 .
Simplify the expression to y = ( c o s x s i n x ) 2 .
Rewrite in terms of tangent to get the final simplified form: y = tan 2 x .
Explanation
Initial Analysis We are given the function y = ( 1 + c o s 2 x s i n 2 x ) 2 and we want to simplify it using trigonometric identities.
Applying Double Angle Identities We will use the double angle identities: sin 2 x = 2 sin x cos x and cos 2 x = 2 cos 2 x − 1 . Substituting these into the function, we get:
y = ( 1 + ( 2 c o s 2 x − 1 ) 2 s i n x c o s x ) 2
Simplifying the Denominator Simplifying the denominator, we have:
y = ( 2 c o s 2 x 2 s i n x c o s x ) 2
Cancelling Common Factors Now, we can cancel out the common factor of 2 cos x :
y = ( c o s x s i n x ) 2
Rewriting in Terms of Tangent Since c o s x s i n x = tan x , we have:
y = tan 2 x
Final Simplification Therefore, the simplified form of the given function is y = tan 2 x .
Examples
In physics, the tangent function and its square appear in various contexts, such as describing the angle of a projectile's trajectory or analyzing the behavior of waves. Simplifying trigonometric expressions like this can make calculations easier and provide a clearer understanding of the underlying relationships. For instance, in optics, the tangent of an angle is used to determine the refractive index of a material, which is crucial for designing lenses and other optical components.
