26.2 Determine the value of each of the following expressions: $$ 26.2 .1 \frac{\cos \left(90^{\circ}-2 heta ight) \cdot \sin heta}{\sin ^{2}\left(180^{\circ}+ heta ight) \cdot \cos \left(720^{\circ}+ heta ight)} $$ $$ 26.2 .2 \frac{1}{\sin ^{2} 2 x}-\frac{1}{ an ^{2} 2 x} $$

Question

26.2 Determine the value of each of the following expressions: $$ 26.2 .1 \frac{\cos \left(90^{\circ}-2 	hetaight) \cdot \sin 	heta}{\sin ^{2}\left(180^{\circ}+	hetaight) \cdot \cos \left(720^{\circ}+	hetaight)} $$ $$ 26.2 .2 \frac{1}{\sin ^{2} 2 x}-\frac{1}{	an ^{2} 2 x} $$

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1. We start with the expression
   $$
   \frac{\cos (90^\circ-2\theta) \cdot \sin \theta}{\sin^2 (180^\circ+\theta) \cdot \cos (720^\circ+\theta)}.
   $$
   
   **Step 1:** Use the co-function identity:
   $$
   \cos (90^\circ-2\theta) = \sin (2\theta).
   $$
   Thus, the numerator becomes $$\sin (2\theta) \cdot \sin \theta$$.

**Step 2:** Simplify the denominator.
   - For $$\sin^2 (180^\circ+\theta)$$: Use the identity $$\sin(180^\circ+\theta) = -\sin \theta$$. Squaring, we have
     $$
     \sin^2 (180^\circ+\theta) = \sin^2 \theta.
     $$
   - For $$\cos (720^\circ+\theta)$$: Since $$720^\circ$$ represents two full rotations, we have
     $$
     \cos (720^\circ+\theta) = \cos \theta.
     $$
   Thus, the denominator becomes $$\sin^2 \theta \cdot \cos \theta$$.

**Step 3:** Substitute the simplified parts into the expression:
   $$
   \frac{\sin (2\theta) \cdot \sin \theta}{\sin^2 \theta \cdot \cos \theta}.
   $$

**Step 4:** Use the double angle formula:
   $$
   \sin(2\theta) = 2\sin \theta \cos \theta.
   $$
   Substituting,
   $$
   \frac{2\sin \theta \cos \theta \cdot \sin \theta}{\sin^2 \theta \cdot \cos \theta} = \frac{2\sin^2 \theta \cos \theta}{\sin^2 \theta \cdot \cos \theta}.
   $$

**Step 5:** Cancel common factors $$\sin^2 \theta$$ and $$\cos \theta$$ (assuming they are not zero), leaving:
   $$
   2.
   $$

Therefore, the value of the expression is $$\boxed{2}$$.

2. Consider the expression
   $$
   \frac{1}{\sin^2 2x} - \frac{1}{\tan^2 2x}.
   $$
   
   **Step 1:** Express $$\tan^2 2x$$ in terms of sine and cosine:
   $$
   \tan^2 2x = \frac{\sin^2 2x}{\cos^2 2x}.
   $$
   Therefore,
   $$
   \frac{1}{\tan^2 2x} = \frac{\cos^2 2x}{\sin^2 2x}.
   $$

**Step 2:** Substitute into the given expression:
   $$
   \frac{1}{\sin^2 2x} - \frac{\cos^2 2x}{\sin^2 2x} = \frac{1 - \cos^2 2x}{\sin^2 2x}.
   $$

**Step 3:** Use the Pythagorean identity:
   $$
   1 - \cos^2 2x = \sin^2 2x.
   $$
   Substituting,
   $$
   \frac{\sin^2 2x}{\sin^2 2x} = 1.
   $$

Therefore, the value of the expression is $$\boxed{1}$$.
The value of the first expression is 2, and the value of the second expression is 1.

26.2 Determine the value of each of the following expressions:

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