Sure, let's start by simplifying the left-hand side of the equation step-by-step.
Consider the given expression:
[tex]\[
\frac{\sin^4 y - \cos^4 y}{\cot y}
\][/tex]
First, note that the numerator [tex]\(\sin^4 y - \cos^4 y\)[/tex] can be factored using the difference of squares formula:
[tex]\[
\sin^4 y - \cos^4 y = (\sin^2 y)^2 - (\cos^2 y)^2 = (\sin^2 y - \cos^2 y)(\sin^2 y + \cos^2 y)
\][/tex]
We know from trigonometric identities that [tex]\(\sin^2 y + \cos^2 y = 1\)[/tex], so the expression simplifies to:
[tex]\[
\sin^4 y - \cos^4 y = (\sin^2 y - \cos^2 y) \cdot 1 = \sin^2 y - \cos^2 y
\][/tex]
Next, let's address the denominator [tex]\(\cot y\)[/tex]. Recall that [tex]\(\cot y = \frac{\cos y}{\sin y}\)[/tex].
Substituting these into our original expression, we get:
[tex]\[
\frac{\sin^2 y - \cos^2 y}{\frac{\cos y}{\sin y}} = (\sin^2 y - \cos^2 y) \cdot \frac{\sin y}{\cos y} = (\sin^2 y - \cos^2 y) \cdot \frac{\sin y}{\cos y}
\][/tex]
This can be then simplified further as:
[tex]\[
(\sin^2 y - \cos^2 y) \cdot \frac{\sin y}{\cos y} = \frac{\sin^3 y}{\cos y} - \frac{\sin y \cos y}{\cos y} = \frac{\sin^3 y}{\cos y} - \sin y
\][/tex]
Which simplifies down to:
[tex]\[
\frac{\sin^3 y}{\cos y} - \sin y = \sin y \cdot \frac{\sin^2 y}{\cos y} - \sin y = \sin y \cdot \frac{\sin^2 y}{\cos y} - \sin y
\][/tex]
Thus:
[tex]\[
\sin y \cdot \tan y - \sin y = \sin y (\tan y - 1) = -\cos(2y) \tan y
\][/tex]
We have:
[tex]\[
\frac{\sin^4 y - \cos^4 y}{\cot y} = -\cos(2y) \tan y
\][/tex]
Now, let's look at the right-hand side of the given equation:
[tex]\[
\tan y - 2 \cos y \sin y
\][/tex]
For a trigonometric identity [tex]\(\cos(2y) = 1 - 2\sin^2 y = \cos^2 x - \sin^2 x \)[/tex], the simplified from:
[tex]\[
\tan y - 2 \sin y \cos y = -\cos(2y) \tan y = -2 \sin y \cos y +\tan y
\][/tex]
So, we have:
[tex]\[
-2 \sin y \cos y + \tan y
\][/tex]
And since [tex]\(\frac{sin^4 y - \cos^4 y}{\cot y} \neq \tan y - 2 \sin y \cos y\)[/tex],
Thus, we have shown that:
[tex]\[
\frac{\sin^4 y - \cos^4 y}{\cot y} \neq \tan y - 2 \cos y \sin y
\][/tex]
So the given equation does not hold.
