Simplifying the Trigonometric Expression: secA * cscA / (1 tanA)

In this article, we will break down and simplify the given trigonometric expression secA * cscA / (1 tanA). This process will involve the use of trigonometric identities and algebraic manipulation. By the end of this article, you'll be able to understand and simplify this expression yourself.

Step-by-Step Solution Explained

Consider the given expression:

secA * cscA / (1 tanA)

Step 1: Express secA and cscA in terms of sinA, cosA

Recall that the secant (secA) and cosecant (cscA) functions can be expressed in terms of sine (sinA) and cosine (cosA):

Substituting these into the original expression, we have:

(1/cosA * 1/sinA) / (1 tanA)

Step 2: Express tanA in terms of sinA and cosA

The tangent (tanA) function is given by:

Substituting this into the expression, the denominator becomes:

1 (sinA / cosA) (cosA sinA) / cosA

Thus, the expression simplifies to:

(1/cosA * 1/sinA) / ((cosA sinA) / cosA)

Step 3: Simplify the expression using algebraic manipulation

Multiplying the numerator and the denominator by cosA, we get:

(cosA / (cosA * sinA)) / ((cosA sinA) / cosA) cosA / (sinA * (cosA sinA))

Note that we can write:

(cosA / (sinA * (cosA sinA))) * (sinA * cosA) / (sinA * cosA) (sinA * cosA) / (sinA * cosA * (cosA sinA)) 1 / (cosA sinA)

Step 4: Verifying the identity

Upon simplifying, we find that the given expression reduces to:

1 / (cosA sinA)

However, by closely examining the initial and final forms, we can see that a more useful and immediate simplification is found by a different approach.

Reevaluation: Simplified Form

Let's revisit the initial expression, directly using the relations:

secA * cscA / (1 tanA) (1/cosA * 1/sinA) / (1 sinA/cosA) (1/cosA * 1/sinA) / ((cosA sinA)/cosA) cosA / (sinA * (cosA sinA)) cscA * (cosA - sinA) / (cosA sinA)

Recognizing that the term (cosA - sinA) / (cosA sinA) simplifies to (1 - tanA) / (1 tanA), we obtain:

cscA * (1 - tanA) / (1 tanA) cscA * cotA

Thus, the given expression simplifies to:

cscA * cotA

Conclusion

Through the steps mentioned above, we simplified the complex trigonometric expression secA * cscA / (1 tanA) to cscA * cotA. This process involved the application of trigonometric identities and algebraic manipulation, showcasing how these techniques can be used to simplify seemingly complex trigonometric expressions.

Understanding these steps not only helps in solving similar problems but also builds a strong foundation in trigonometry. Equations and expressions become less daunting when broken down using these fundamental principles.