Step-by-step Solution:

\[ \cot^2 45^\circ - \sin^2 45^\circ = K \sin^2 30^\circ \times \tan^2 45^\circ \times \sec^2 45^\circ \] Step 1: Evaluate Trigonometric Values We first compute the required trigonometric values: \[ \cot 45^\circ = 1, \quad \sin 45^\circ = \frac{1}{\sqrt{2}}, \quad \sin 30^\circ = \frac{1}{2} \] \[ \tan 45^\circ = 1, \quad \sec 45^\circ = \sqrt{2} \] Step 2: Compute Left-Hand Side \[ \cot^2 45^\circ = (1)^2 = 1 \] \[ \sin^2 45^\circ = \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{2} \] \[ \cot^2 45^\circ - \sin^2 45^\circ = 1 - \frac{1}{2} = \frac{1}{2} \] Step 3: Compute Right-Hand Side \[ \sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \] \[ \tan^2 45^\circ = (1)^2 = 1 \] \[ \sec^2 45^\circ = (\sqrt{2})^2 = 2 \] \[ K \times \frac{1}{4} \times 1 \times 2 = K \times \frac{2}{4} = K \times \frac{1}{2} \] Step 4: Solve for \( K \) \[ \frac{1}{2} = K \times \frac{1}{2} \] \[ K = 1 \] Final Answer \[ \boxed{1} \]