Questions: Simplify (sin(t))^2 / ((sin(t))^2 + (cos(t))^2) to an expression involving a single trig function with no fractions. If needed, enter squared trigonometric expressions using the following notation. Example: Enter (sin(t))^2 as (sin(t))^2.

Solution Steps

To simplify the given expression, we can use the Pythagorean identity for trigonometric functions, which states that \(\sin^2(t) + \cos^2(t) = 1\). This identity allows us to simplify the denominator of the given fraction.

1. Recognize and apply the Pythagorean identity \(\sin^2(t) + \cos^2(t) = 1\).
2. Substitute the identity into the denominator of the given fraction.
3. Simplify the resulting expression.

We start with the expression

\[ \frac{\sin^2(t)}{\sin^2(t) + \cos^2(t)}. \]

Using the Pythagorean identity, we know that

\[ \sin^2(t) + \cos^2(t) = 1. \]

Substituting the identity into the denominator, we have:

\[ \frac{\sin^2(t)}{1}. \]

This simplifies directly to

\[ \sin^2(t). \]

Final Answer