$y=\left(\frac{\sin 2 x}{1+\cos 2 x}\right)^2$
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 these formulas into the given expression.
Simplify the resulting fraction.
Recognize the tangent identity tan x = c o s x s i n x to obtain the final simplified form: 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. We will use trigonometric identities to rewrite the expression in a simpler form.
Double Angle Formulas We know the double angle formulas:
Double Angle Formulas sin 2 x = 2 sin x cos x and cos 2 x = 2 cos 2 x − 1 .
Substitution Substitute these formulas into the expression:
Substitution y = ( 1 + ( 2 c o s 2 x − 1 ) 2 s i n x c o s x ) 2 = ( 2 c o s 2 x 2 s i n x c o s x ) 2
Simplification Simplify the fraction inside the parentheses:
Simplification y = ( c o s x s i n x ) 2
Tangent Identity Recognize that c o s x s i n x = tan x , so
Tangent Identity y = ( tan x ) 2 = tan 2 x
Final Answer Therefore, the simplified form of the given expression is tan 2 x .
Examples
Imagine you're designing a solar panel tracking system. The amount of sunlight captured changes with the angle of the sun, and this angle can be described using trigonometric functions. Simplifying trigonometric expressions, like we did here, helps in optimizing the panel's orientation to maximize energy capture throughout the day. This ensures efficient energy conversion and reduces energy waste.
