Hey guys! Let's dive into simplifying the trigonometric expression oscsinacosbsc + sccosa sinbsc. This might look a bit intimidating at first glance, but don't worry! We'll break it down step by step to make it super easy to understand. Trigonometric simplification is a fundamental skill in mathematics, especially when you're dealing with calculus, physics, or engineering problems. Mastering these simplifications can save you a ton of time and reduce errors. Think of it as leveling up your math game! So, let's get started and turn this complex expression into something much more manageable. The beauty of trigonometry lies in its interconnectedness. By understanding basic identities and relationships, seemingly complex expressions can be simplified into elegant forms. This not only makes problem-solving easier but also reveals underlying symmetries and patterns. As we proceed, remember that each step builds upon the previous one. Make sure you grasp the core concepts before moving on. Okay, let's begin with understanding each of the terms within the expression. We need to identify what each term represents and how they relate to one another. Remember, practice makes perfect, so don't hesitate to revisit these concepts and work through similar examples. This will help solidify your understanding and build your confidence in tackling more challenging problems.

Understanding the Expression

Before we jump into simplifying, let's clarify what each part of the expression oscsinacosbsc + sccosa sinbsc means. Breaking down the terms is essential for a clear understanding of the problem. When we understand the components, it becomes easier to identify the strategies we can use to make simplification more efficient.

• osc: This seems like a typo and should probably be interpreted in the context of standard trigonometric functions. There is no standard trigonometric function denoted as osc. Instead, it is most likely referring to csc, which is the cosecant function. The cosecant function is defined as the reciprocal of the sine function, that is, csc(x) = 1/sin(x).
• sina: This is the sine function of angle a, written as sin(a).
• cosb: This is the cosine function of angle b, written as cos(b).
• sc: Again, this seems like a typo. There is no standard trigonometric function sc. In all likelihood, this refers to sec, which stands for the secant function. The secant function is the reciprocal of the cosine function, that is, sec(x) = 1/cos(x).
• cosa: This is the cosine function of angle a, written as cos(a).
• sinb: This is the sine function of angle b, written as sin(b).

Given the likelihood of typos, the expression probably intends to be csc(a)cos(b) + sec(a)sin(b). If we proceed with this interpretation, the simplification becomes much clearer. This adjustment relies on recognizing patterns and common notations within trigonometry. It's a skill that sharpens with experience and familiarity. So, assuming the original expression contains those typos it becomes more clear what we should be simplifying. If you come across similar expressions, make sure you double-check to ensure that the terms are properly notated. Now, let's assume that we have correctly interpreted the expression and move on to simplification. It involves applying trigonometric identities and algebraic manipulations.

Rewrite the Expression with Corrected Terms

Okay, let's rewrite the expression with the corrected terms. Assuming that osc was meant to be csc (cosecant) and sc was meant to be sec (secant), our expression now looks like this:

csc(a)cos(b) + sec(a)sin(b)

Now that we have what seems to be the correct expression, let's express csc(a) and sec(a) in terms of sine and cosine to make it easier to simplify. This involves using reciprocal identities, which are fundamental in trigonometry. These identities allow us to rewrite trigonometric functions in terms of their reciprocals, making simplification easier. By expressing everything in terms of sines and cosines, we reduce the complexity of the expression and bring it down to its most fundamental components. Okay, let's jump to rewriting the expression with the new substitutions. This will make our next steps much clearer and more straightforward.

• Recall that csc(a) = 1/sin(a) and sec(a) = 1/cos(a).

So, we can substitute these into our expression:

(1/sin(a))cos(b) + (1/cos(a))sin(b)

Now the expression looks like a sum of two fractions. To combine these fractions, we need a common denominator. The common denominator in this case will be sin(a)cos(a). So, let's rewrite each fraction with this common denominator. This step is crucial because it allows us to combine the terms and move towards further simplification. Ensuring that you have the correct common denominator is essential to avoid errors in the subsequent steps. So, let's rewrite the fractions.

Finding a Common Denominator

To combine the two terms, (1/sin(a))cos(b) + (1/cos(a))sin(b), we need to find a common denominator. The common denominator here is simply the product of the two denominators, which is sin(a)cos(a). Now, we'll rewrite each fraction with this common denominator. This involves multiplying the numerator and denominator of each fraction by the appropriate term to achieve the common denominator. It's a fundamental algebraic technique that's widely used in simplifying expressions. So, let's perform the necessary steps.

• For the first term, (1/sin(a))cos(b), we multiply both the numerator and denominator by cos(a): (cos(b) * cos(a)) / (sin(a) * cos(a)) = cos(a)cos(b) / (sin(a)cos(a))
• For the second term, (1/cos(a))sin(b), we multiply both the numerator and denominator by sin(a): (sin(b) * sin(a)) / (cos(a) * sin(a)) = sin(a)sin(b) / (sin(a)cos(a))

Now we have a common denominator, so we can combine the two fractions:

(cos(a)cos(b) + sin(a)sin(b)) / (sin(a)cos(a))

Now we can see a familiar pattern in the numerator. Recognizing this pattern is key to simplifying the expression further. The numerator looks very similar to a trigonometric identity that we know very well. Let's take a look at that identity and see if it applies to our case.

Applying the Cosine Addition Identity

Do you notice anything familiar about the numerator? The numerator, cos(a)cos(b) + sin(a)sin(b), is actually the expansion of the cosine subtraction identity! Remember that:

cos(a - b) = cos(a)cos(b) + sin(a)sin(b)

So, we can replace the numerator with cos(a - b):

cos(a - b) / (sin(a)cos(a))

Now our expression looks much simpler. But we can simplify it even further! Let's focus on the denominator, sin(a)cos(a). It looks very similar to another trigonometric identity, the double angle identity for sine. Recognizing and applying these identities is key to mastering trigonometric simplifications. With a little more manipulation, we can achieve an even simpler result.

Using the Double Angle Identity for Sine

Now, let's turn our attention to the denominator, which is sin(a)cos(a). Recall the double angle identity for sine:

sin(2a) = 2sin(a)cos(a)

We can rewrite our denominator in terms of sin(2a) as follows:

sin(a)cos(a) = (1/2)sin(2a)

Now, substitute this back into our expression:

cos(a - b) / ((1/2)sin(2a))

To get rid of the fraction in the denominator, we can multiply both the numerator and the denominator by 2:

(2cos(a - b)) / sin(2a)

Final Simplified Expression

Alright, guys, we've successfully simplified the expression! After all those steps, our final simplified form is:

(2cos(a - b)) / sin(2a)

This is the simplified form of the original expression oscsinacosbsc + sccosa sinbsc (after correcting the typos). Remember, the key steps were:

• Correcting the typos in the original expression.
• Expressing cosecant and secant in terms of sine and cosine.
• Finding a common denominator.
• Applying the cosine subtraction identity.
• Using the double angle identity for sine.

By following these steps, you can tackle similar trigonometric simplification problems. Keep practicing, and you'll become a pro in no time! Remember, understanding the underlying principles and trigonometric identities is the key to success in these problems. Don't be afraid to take your time and carefully go through each step to ensure accuracy. Happy simplifying!