5.3 Simplify each expression to a single term: $$ 5.5 .1 \quad \cos ^{2} 19^{\circ}-\sin ^{2} 19^{\circ}= $$ $$ \qquad $$ $$ 5.5 .2 \cos ^{2} .19^{\circ}+\sin ^{2} 19^{\circ}= $$ $$ \qquad $$ $$ 5.5 .3 \quad 2 \cos ^{2} 50^{\circ}-1= $$ $$ \qquad $$ $$ 5.5 .4 . \quad 1-2 \sin ^{2} 50^{\circ}= $$ $$ \qquad $$ $$ 5.5 .5 \quad 6 \cos ^{2} \frac{ heta}{2}-3= $$ $$ \qquad $$

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5.3 Simplify each expression to a single term: $$ 5.5 .1 \quad \cos ^{2} 19^{\circ}-\sin ^{2} 19^{\circ}= $$ $$ \qquad $$ $$ 5.5 .2 \cos ^{2} .19^{\circ}+\sin ^{2} 19^{\circ}= $$ $$ \qquad $$ $$ 5.5 .3 \quad 2 \cos ^{2} 50^{\circ}-1= $$ $$ \qquad $$ $$ 5.5 .4 . \quad 1-2 \sin ^{2} 50^{\circ}= $$ $$ \qquad $$ $$ 5.5 .5 \quad 6 \cos ^{2} \frac{	heta}{2}-3= $$ $$ \qquad $$

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1. $$ \cos^2 19^\circ - \sin^2 19^\circ $$  
   Using the double-angle identity  
   $$
   \cos 2\theta = \cos^2\theta - \sin^2\theta,
   $$  
   let $$\theta = 19^\circ$$. This yields  
   $$
   \cos^2 19^\circ - \sin^2 19^\circ = \cos 38^\circ.
   $$

2. $$ \cos^2 19^\circ + \sin^2 19^\circ $$  
   Using the Pythagorean identity  
   $$
   \cos^2\theta + \sin^2\theta = 1,
   $$  
   we have  
   $$
   \cos^2 19^\circ + \sin^2 19^\circ = 1.
   $$

3. $$ 2\cos^2 50^\circ - 1 $$  
   Using the double-angle identity for cosine  
   $$
   \cos 2\theta = 2\cos^2\theta - 1,
   $$  
   let $$\theta = 50^\circ$$. Thus,  
   $$
   2\cos^2 50^\circ - 1 = \cos 100^\circ.
   $$

4. $$ 1 - 2\sin^2 50^\circ $$  
   Using the alternative double-angle identity  
   $$
   \cos 2\theta = 1 - 2\sin^2\theta,
   $$  
   let $$\theta = 50^\circ$$. This gives  
   $$
   1 - 2\sin^2 50^\circ = \cos 100^\circ.
   $$

5. $$ 6\cos^2 \frac{\theta}{2} - 3 $$  
   Factor out 3 from the expression:  
   $$
   6\cos^2 \frac{\theta}{2} - 3 = 3\left(2\cos^2 \frac{\theta}{2} - 1\right).
   $$  
   Using the double-angle identity  
   $$
   \cos\theta = 2\cos^2 \frac{\theta}{2} - 1,
   $$  
   we obtain  
   $$
   6\cos^2 \frac{\theta}{2} - 3 = 3\cos\theta.
   $$
1. $$ \cos 38^\circ $$ 2. $$ 1 $$ 3. $$ \cos 100^\circ $$ 4. $$ \cos 100^\circ $$ 5. $$ 3\cos\theta $$

5.3 Simplify each expression to a single term:

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