Question

Find functions \(f\) and \(g\) such that \(h=g \circ f\) (Note: The answer is not unique.) \(h(t)=\frac{\tan t}{1+\cot t}\)

Short answer

We can choose \(f(t) = \sin t\) and \(g(t) = \frac{1}{\cos t + \sin t}\). These functions satisfy the composition property \(h(t) = (g \circ f)(t) = g(f(t)) = \frac{\sin t}{\cos t + \sin t}\).

Step-by-step solution

Rewrite h(t) in terms of sine and cosine

We notice that the given function \(h(t)\) is in terms of tangent and cotangent functions. First, let's convert the cotangent function into a reciprocal form of the tangent function: $$ h(t)=\frac{\tan t}{1+\frac{1}{\tan t}} $$ Now, rewrite the tangent function in terms of sine and cosine: $$ h(t)=\frac{\frac{\sin t}{\cos t}}{1+\frac{1}{\frac{\sin t}{\cos t}}} $$ Simplify the expression: $$ h(t)=\frac{\frac{\sin t}{\cos t}}{1+\frac{\cos t}{\sin t}} $$ Multiply both numerator and denominator by \(\cos t\) to eliminate the fractions inside the expressions: $$ h(t)=\frac{\sin t}{\cos t + \sin t} $$ Now we have \(h(t)\) in a more straightforward form.

Choose f and g functions

Since we have simplified the expression for \(h(t)\), we can now look for a natural and easily recognizable split between functions \(f(t)\) and \(g(t)\). The most obvious split would be on the numerator and denominator of the fraction: Let's choose \(f(t) = \sin t\) and \(g(t) = \frac{1}{\cos t + \sin t}\).

Verify the composition of functions

Now that we have chosen two functions, let's verify that \(g \circ f\) gives us the original \(h(t)\) function: $$ (g\circ f)(t) = g\left(f(t)\right) = g(\sin t) $$ Plugging the \(\sin t\) into the function \(g(t)\): $$ g(\sin t) = \frac{1}{\cos t + \sin t} $$ Finally, multiply by \(f(t)\): $$ h(t) = \sin t \cdot \frac{1}{\cos t + \sin t} = \frac{\sin t}{\cos t + \sin t} $$ This matches our simplified form of \(h(t)\), so our choice of functions \(f(t)\) and \(g(t)\) was successful.

Key concepts

Trigonometric Identities

Understanding trigonometric identities is vital for simplifying complex expressions involving trigonometric functions. These identities are equations that relate the six fundamental trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. They are true for all values within the functions' domains. Basic trigonometric identities include reciprocal identities like \( \cot t = \frac{1}{\tan t} \) and Pythagorean identities such as \(\tan^2 t + 1 = \( \frac{1}{\tan t} \)\^2\).

In our problem, converting \(\tan t\) and \(\frac{1}{\tan t}\), which is \(\frac{\tan t}{1+\frac{1}{\tan t}}\), is an application of trigonometric identities. They help us rewrite the function in terms of sine and cosine, which are \(\frac{\frac{\/sin t}{\cos t}}{1+\frac{\cos t}{\/sin t}}\). Recognizing and applying these identities correctly leads to a simplified form of the function that is easier to work with when we're trying to compose functions.

Function Simplification

Simplifying trigonometric functions is similar to simplifying any algebraic expression: it involves factoring, expanding, cancelling out terms, and using identities to rewrite the functions in simpler forms. The goal is to transform the function into a form that is easier to interpret or use in further operations, like function composition.

In the solution presented, the initial complex fraction involving tangent and cotangent is simplified to \(\frac{\sin t}{\cos t + \sin t}\) by multiplying both numerator and denominator by \(\cos t\), which effectively removes the complex fraction and leaves us with a single fraction that is far easier to decompose into two functions, f(t) and g(t). This skillful manipulation showcases function simplification.

Verifying Function Composition

Finally, the composition of functions, often denoted by \(g \circ f\), involves applying one function to the result of another. Verifying that we have the correct composition is as important as finding the functions themselves. To ensure that our composed function \(g \circ f(t)\) produces the original function \(h(t)\), we substitute \(f(t)\) into \(g(t)\) and perform the operations to see if the outcome matches \(h(t)\).

In our case, we start with \(f(t) = \sin t\) and \(g(t) = \frac{1}{\cos t + \sin t}\). Substituting \(\sin t\) into \(g(t)\) and then multiplying by \(f(t)\) verifies the composition because it yields the simplified function \(h(t) = \frac{\sin t}{\cos t + \sin t}\), which matches our target function. This step is crucial for proving our selected functions \(f\) and \(g\) are correct in the composition.