$y=\left(\frac{\sin 2 x}{1+\cos 2 x}\right)^2$
Answer (1)
Use the double angle formulas sin 2 x = 2 sin x cos x and cos 2 x = 2 cos 2 x − 1 .
Substitute into the expression: y = ( 1 + ( 2 c o s 2 x − 1 ) 2 s i n x c o s x ) 2 .
Simplify to y = ( c o s x s i n x ) 2 .
Rewrite using the tangent function: y = tan 2 x .
Explanation
Problem Analysis We are given the function y = ( 1 + c o s 2 x s i n 2 x ) 2 and asked to simplify it using trigonometric identities.
Double Angle Formulas We will use the double angle formulas to rewrite the expression. Recall that sin 2 x = 2 sin x cos x and cos 2 x = 2 cos 2 x − 1 .
Substitution Substitute these formulas into the given expression:
y = ( 1 + ( 2 c o s 2 x − 1 ) 2 s i n x c o s x ) 2
Simplification Simplify the expression:
y = ( 2 c o s 2 x 2 s i n x c o s x ) 2
Cancellation Cancel out the common factors:
y = ( c o s x s i n x ) 2
Tangent Definition Rewrite the expression using the definition of tangent: tan x = c o s x s i n x . Therefore,
y = tan 2 x
Final Answer Thus, the simplified form of the given expression is tan 2 x .
Examples
Imagine you're designing a solar panel system. The amount of sunlight a panel captures can be modeled using trigonometric functions. Simplifying these functions, like we did with the double angle formula, helps optimize the panel's angle for maximum energy capture. This ensures efficient energy production throughout the day, saving resources and maximizing output.
