Trigonometric Identities

Step 1: Concept
Trigonometric identity simplification and rationalization.
Step 2: Analysis

a%20%3D%20%5Cfrac%7B2%20%5Csin%20%5Ctheta%7D%7B1%20%2B%20%5Csin%20%5Ctheta%20%2B%20%5Ccos%20%5Ctheta%7D

.
Multiply numerator and denominator by

(1%20%2B%20%5Csin%20%5Ctheta%20-%20%5Ccos%20%5Ctheta)

a%20%3D%20%5Cfrac%7B2%20%5Csin%20%5Ctheta%20(1%20%2B%20%5Csin%20%5Ctheta%20-%20%5Ccos%20%5Ctheta)%7D%7B(1%20%2B%20%5Csin%20%5Ctheta)%5E2%20-%20%5Ccos%5E2%20%5Ctheta%7D

Step 3: Calculation
Denominator

%3D%201%20%2B%20%5Csin%5E2%20%5Ctheta%20%2B%202%20%5Csin%20%5Ctheta%20-%20%5Ccos%5E2%20%5Ctheta

%3D%20(1%20-%20%5Ccos%5E2%20%5Ctheta)%20%2B%20%5Csin%5E2%20%5Ctheta%20%2B%202%20%5Csin%20%5Ctheta

%3D%20%5Csin%5E2%20%5Ctheta%20%2B%20%5Csin%5E2%20%5Ctheta%20%2B%202%20%5Csin%20%5Ctheta%20%3D%202%20%5Csin%5E2%20%5Ctheta%20%2B%202%20%5Csin%20%5Ctheta%20%3D%202%20%5Csin%20%5Ctheta%20(1%20%2B%20%5Csin%20%5Ctheta)

.
So,

a%20%3D%20%5Cfrac%7B2%20%5Csin%20%5Ctheta%20(1%20%2B%20%5Csin%20%5Ctheta%20-%20%5Ccos%20%5Ctheta)%7D%7B2%20%5Csin%20%5Ctheta%20(1%20%2B%20%5Csin%20%5Ctheta)%7D

a%20%3D%20%5Cfrac%7B1%20%2B%20%5Csin%20%5Ctheta%20-%20%5Ccos%20%5Ctheta%7D%7B1%20%2B%20%5Csin%20%5Ctheta%7D

.
Step 4: Conclusion
The value is equal to

a

.
Final Answer:(C)

To solve for $%5Cfrac%7Bdy%7D%7Bdx%7D$ given that $(%5Ccos%20x)%5Ey%20%3D%20(%5Csin%20y)%5Ex$ , we start by taking the natural logarithm on both sides of the equation:

$%5Cln((%5Ccos%20x)%5Ey)%20%3D%20%5Cln((%5Csin%20y)%5Ex)$

This gives:

$y%20%5Cln(%5Ccos%20x)%20%3D%20x%20%5Cln(%5Csin%20y)$

Now, differentiate both sides with respect to $x$ . For the left side, apply the product rule:

$%5Cfrac%7Bd%7D%7Bdx%7D(y%20%5Cln(%5Ccos%20x))%20%3D%20y%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D(%5Cln(%5Ccos%20x))%20%2B%20%5Cln(%5Ccos%20x)%20%5Ccdot%20%5Cfrac%7Bdy%7D%7Bdx%7D$

$%5Cfrac%7Bd%7D%7Bdx%7D(%5Cln(%5Ccos%20x))%20%3D%20%5Cfrac%7B1%7D%7B%5Ccos%20x%7D%20%5Ccdot%20(-%5Csin%20x)%20%3D%20-%5Ctan%20x$

Thus, the derivative of the left side is:

$y%20(-%5Ctan%20x)%20%2B%20%5Cln(%5Ccos%20x)%20%5Ccdot%20%5Cfrac%7Bdy%7D%7Bdx%7D$

For the right side, apply the product rule:

$%5Cfrac%7Bd%7D%7Bdx%7D(x%20%5Cln(%5Csin%20y))%20%3D%20x%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D(%5Cln(%5Csin%20y))%20%2B%20%5Cln(%5Csin%20y)%20%5Ccdot%20%5Cfrac%7Bd%7D%7Bdx%7D(x)$

$%5Cfrac%7Bd%7D%7Bdx%7D(%5Cln(%5Csin%20y))%20%3D%20%5Cfrac%7B1%7D%7B%5Csin%20y%7D%5Ccdot%5Ccos%20y%5Ccdot%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Ccot%20y%20%5Ccdot%20%5Cfrac%7Bdy%7D%7Bdx%7D$

The derivative of the right side is:

$x%20%5Ccdot%20%5Ccot%20y%20%5Ccdot%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%2B%20%5Cln(%5Csin%20y)$

Equate the derivatives:

$y%20(-%5Ctan%20x)%20%2B%20%5Cln(%5Ccos%20x)%20%5Ccdot%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20x%20%5Ccdot%20%5Ccot%20y%20%5Ccdot%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%2B%20%5Cln(%5Csin%20y)$

Rearrange to solve for $%5Cfrac%7Bdy%7D%7Bdx%7D$ :

$%5Cln(%5Ccos%20x)%20%5Ccdot%20%5Cfrac%7Bdy%7D%7Bdx%7D%20-%20x%20%5Ccdot%20%5Ccot%20y%20%5Ccdot%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cln(%5Csin%20y)%20%2B%20y%20%5Ctan%20x$

Factor out $%5Cfrac%7Bdy%7D%7Bdx%7D$ :

$%5Cfrac%7Bdy%7D%7Bdx%7D(%5Cln(%5Ccos%20x)%20-%20x%20%5Ccdot%20%5Ccot%20y)%20%3D%20%5Cln(%5Csin%20y)%20%2B%20y%20%5Ctan%20x$

Solve for $%5Cfrac%7Bdy%7D%7Bdx%7D$ :

$%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cfrac%7B%5Cln(%5Csin%20y)%20%2B%20y%20%5Ctan%20x%7D%7B%5Cln(%5Ccos%20x)%20-%20x%20%5Ccdot%20%5Ccot%20y%7D$

Thus, the solution is:

$%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%20%5Cfrac%7B%5Clog_e%20%5Csin%20y%20-%20y%20%5Ctan%20x%7D%7B%5Clog_e%20%5Ccos%20x%20%2B%20x%20%5Ccot%20y%7D$

About Trigonometric Identities - CUET-UG

Trigonometric Identities is a vital chapter for CUET-UG aspirants. Mastering the concepts covered in this chapter is essential for securing a top rank.

By rigorously practicing the previous year questions associated with this chapter, you can identify high-yield topics, understand the examiner's perspective, and boost your confidence during the actual exam.

Frequently Asked Questions

Why focus on Trigonometric Identities PYQs?

Analyzing PYQs for this specific chapter reveals the most frequently tested concepts and the typical complexity of questions, allowing you to tailor your study plan efficiently.

How to best use this analysis?

Review the topic breakdown to see which sub-topics within Trigonometric Identities carry the most weight. Then, tackle the questions iteratively to solidify your understanding.

High-Yield Trend

Chapter Questions
3 MCQs

Official Solution

Official Solution

Official Solution

About Trigonometric Identities - CUET-UG

Frequently Asked Questions

Why focus on Trigonometric Identities PYQs?

How to best use this analysis?

Trigonometric Identities

High-Yield Trend

Chapter Questions 3 MCQs

Official Solution

Official Solution

Official Solution

About Trigonometric Identities - CUET-UG

Frequently Asked Questions

Why focus on Trigonometric Identities PYQs?

How to best use this analysis?

Chapter Questions
3 MCQs