https://file.aiquickdraw.com/mathbot/file/1728877465071qcmzk3me.jpg

Let's go through the steps to simplify each expression:

### (a) \(\frac{\cos \theta}{\sin(90^\circ - \theta)}\):

We can use the trigonometric identity:
\[
\sin(90^\circ - \theta) = \cos \theta
\]
So, the expression becomes:
\[
\frac{\cos \theta}{\sin(90^\circ - \theta)} = \frac{\cos \theta}{\cos \theta}
\]

Since \(\cos \theta\) cancels out (assuming \(\theta \neq 90^\circ\)), the expression simplifies to:
\[
1
\]

### (b) \(\cos(90^\circ + \theta) + \sin \theta\):

We use the identity:
\[
\cos(90^\circ + \theta) = -\sin \theta
\]
So, the expression becomes:
\[
\cos(90^\circ + \theta) + \sin \theta = -\sin \theta + \sin \theta
\]

The terms cancel out, simplifying to:
\[
0
\]

### Final Answers:
- (a) simplifies to **1**.
- (b) simplifies to **0**.

Would you like further details or explanations on any of these steps?

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Here are 5 related questions you can explore:
1. What is the proof of the identity \(\sin(90^\circ - \theta) = \cos \theta\)?
2. How do the co-function identities help simplify trigonometric expressions?
3. Can \(\sin(90^\circ + \theta)\) also be simplified using similar rules?
4. What other trigonometric identities can be derived from angles related to \(90^\circ\)?
5. How would the simplifications change if we use radians instead of degrees?

**Tip:** Knowing the basic trigonometric identities for complementary and supplementary angles makes simplifying such expressions much quicker.

Simplify: (a) cos(θ) / sin(90° - θ) (b) cos(90° + θ) + sin(θ)

Learn how to simplify trigonometric expressions like cos(θ) / sin(90° - θ) and cos(90° + θ) + sin(θ) using co-function identities and angle sum identities. Suitable for high school students studying trigonometry.

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