Let's solve the problems from the image step by step.

Problem 5:

Given $\sin(\theta) = \frac{3}{5}$ and $\theta$ is in the second quadrant, find:
(a) $\cos(\theta)$, (b) $\tan(\theta)$, (c) $\sec(\theta)$.

Solution:

1. Use the Pythagorean identity:
   $\sin^2(\theta) + \cos^2(\theta) = 1$
   Substitute $\sin(\theta) = \frac{3}{5}$:
   $\left(\frac{3}{5}\right)^2 + \cos^2(\theta) = 1$
   $\frac{9}{25} + \cos^2(\theta) = 1$
   $\cos^2(\theta) = 1 - \frac{9}{25} = \frac{16}{25}$
   $\cos(\theta) = \pm \frac{4}{5}$
   Since $\theta$ is in the second quadrant, $\cos(\theta) < 0$:
   $\cos(\theta) = -\frac{4}{5}.$

2. Find $\tan(\theta)$:
   $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{\frac{3}{5}}{-\frac{4}{5}} = -\frac{3}{4}.$

3. Find $\sec(\theta)$:
   $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{-\frac{4}{5}} = -\frac{5}{4}.$

Final Answers for Problem 5:

(a) $\cos(\theta) = -\frac{4}{5}$
(b) $\tan(\theta) = -\frac{3}{4}$
(c) $\sec(\theta) = -\frac{5}{4}$

Problem 6:

Simplify $\sin(2x) + \cos(2x)$ using double-angle identities.

Solution:

1. Recall the double-angle identities:
   $\sin(2x) = 2\sin(x)\cos(x), \quad \cos(2x) = \cos^2(x) - \sin^2(x).$

2. Substitute into the given expression:
   $\sin(2x) + \cos(2x) = 2\sin(x)\cos(x) + (\cos^2(x) - \sin^2(x)).$

3. Rearrange and simplify:
   Combine $\cos^2(x) - \sin^2(x) + 2\sin(x)\cos(x)$:
   This becomes:
   $(\cos(x) + \sin(x))^2 - 1$
   Using the square expansion and simplifications.

4. The simplified form is:
   $(\cos(x) + \sin(x))^2 - 1.$

Would you like detailed steps on Problem 6 or help with any concept?

Related Questions:

1. How do trigonometric functions change in other quadrants?
2. What are alternative forms of double-angle identities?
3. Why does the sign of $\cos(\theta)$ change in the second quadrant?
4. How can $(\cos(x) + \sin(x))^2$ be expanded explicitly?
5. How do you derive double-angle identities from basic trigonometric principles?

Tip:

Always check the quadrant of the angle to determine the correct sign of trigonometric values.