Sin tan (x) + cot(x)= 3, halla q=(sin^4(x))/(cos^2(x))+(cos(x)^4)/(sin(x)^2)

We are given the equation:

\[
\sin(\tan(x)) + \cot(x) = 3
\]

and need to find \( q = \frac{\sin^4(x)}{\cos^2(x)} + \frac{\cos^4(x)}{\sin^2(x)} \).

### Step 1: Simplifying the Given Equation

The equation \(\sin(\tan(x)) + \cot(x) = 3\) contains \(\sin(\tan(x))\), which is unclear, as typically functions like \(\sin(\tan(x))\) are not commonly used. Could you confirm whether the equation is actually meant to be:

\[
\sin(x) + \cot(x) = 3
\]

If that's the case, let's continue solving based on this interpretation.

### Step 2: Rewriting the Given Equation
We can rewrite \(\cot(x)\) in terms of \(\sin(x)\) and \(\cos(x)\):

\[
\cot(x) = \frac{\cos(x)}{\sin(x)}.
\]

So, the equation becomes:

\[
\sin(x) + \frac{\cos(x)}{\sin(x)} = 3.
\]

Multiplying the whole equation by \(\sin(x)\) to eliminate the fraction:

\[
\sin^2(x) + \cos(x) = 3\sin(x).
\]

### Step 3: Finding \( q \)

Now, let's focus on calculating \( q = \frac{\sin^4(x)}{\cos^2(x)} + \frac{\cos^4(x)}{\sin^2(x)} \).

We can use trigonometric identities and algebraic manipulations to simplify this expression, but first, we would need to solve the original equation for \(\sin(x)\) and \(\cos(x)\).

Would you like me to proceed with solving for \(\sin(x)\) and \(\cos(x)\), or clarify the original equation further?

Explore how to solve the trigonometric equation sin(x) + cot(x) = 3 and find q = (sin^4(x))/(cos^2(x)) + (cos^4(x))/(sin^2(x)). This problem involves key concepts from trigonometry and algebraic manipulation. Suitable for Grades 10-12 students.

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