42. Simplify:
sin A/(1 + cos A) + (1 + cos A)/sin A
(A) 2 sec A
(B) 2 cosec A
(C) sec A
(D) cosec A
Simplify sin A/(1 + cos A) + (1 + cos A)/sin A
Understanding the Given Trigonometric Expression
This question is based on the simplification of a trigonometric expression using algebraic operations and fundamental trigonometric identities. The given expression contains two fractions involving sin A and cos A, and our objective is to combine these fractions and reduce the result to one of the standard trigonometric functions given in the options.
The expression to be simplified is:
sin A/(1 + cos A) + (1 + cos A)/sin A
Since the denominators of the two fractions are different, the most direct method is to take a common denominator. After combining the fractions, we can use the fundamental identity sin² A + cos² A = 1 to simplify the numerator.
Taking the Common Denominator
The two fractions in the given expression are:
sin A/(1 + cos A)
and:
(1 + cos A)/sin A
The common denominator is:
sin A(1 + cos A)
Therefore, combining the two fractions gives:
[sin² A + (1 + cos A)²]/[sin A(1 + cos A)]
The expression has now been converted into a single fraction. The next step is to simplify its numerator.
Expanding the Numerator
The numerator of the combined fraction is:
sin² A + (1 + cos A)²
Using the algebraic identity:
(a + b)² = a² + 2ab + b²
we expand (1 + cos A)² as:
(1 + cos A)² = 1 + 2 cos A + cos² A
Therefore, the numerator becomes:
sin² A + 1 + 2 cos A + cos² A
Rearranging the terms:
(sin² A + cos² A) + 1 + 2 cos A
Using the Fundamental Trigonometric Identity
One of the most important identities in trigonometry is:
sin² A + cos² A = 1
Substituting this identity into the numerator gives:
1 + 1 + 2 cos A
Therefore:
2 + 2 cos A
Taking 2 as a common factor:
2(1 + cos A)
Hence, the original expression becomes:
2(1 + cos A)/[sin A(1 + cos A)]
Cancelling the Common Factor
The factor (1 + cos A) appears in both the numerator and denominator. Therefore, it can be cancelled, provided the original expression is defined.
After cancellation, we get:
2/sin A
Now, using the reciprocal trigonometric identity:
cosec A = 1/sin A
we obtain:
2/sin A = 2 cosec A
Therefore, the simplified value of the given expression is:
2 cosec A
Alternative Method Using Rationalization
The same expression can also be simplified by rationalizing the first fraction. Consider:
sin A/(1 + cos A)
Multiply both the numerator and denominator by (1 − cos A):
[sin A(1 − cos A)]/[(1 + cos A)(1 − cos A)]
Using the difference of squares:
(1 + cos A)(1 − cos A) = 1 − cos² A
and the identity:
1 − cos² A = sin² A
the first fraction becomes:
[sin A(1 − cos A)]/sin² A
Cancelling one factor of sin A gives:
(1 − cos A)/sin A
Therefore, the original expression can be written as:
(1 − cos A)/sin A + (1 + cos A)/sin A
Since both fractions now have the same denominator, we combine them:
[(1 − cos A) + (1 + cos A)]/sin A
The terms −cos A and +cos A cancel, giving:
2/sin A
Using cosec A = 1/sin A, we again obtain:
2 cosec A
This alternative method confirms the result obtained by taking the common denominator.
Analysis of All the Given Options
Option (A): 2 sec A
This option is incorrect. The secant function is the reciprocal of cosine, that is, sec A = 1/cos A. However, after complete simplification, the denominator of the expression is sin A, not cos A. Therefore, the result cannot be 2 sec A.
Option (B): 2 cosec A
This option is correct. After taking the common denominator and applying the identity sin² A + cos² A = 1, the expression simplifies to 2/sin A. Since 1/sin A = cosec A, the final result is 2 cosec A.
Option (C): sec A
This option is incorrect because the simplified expression does not contain 1/cos A. The calculation produces 2/sin A, so neither the trigonometric function nor the numerical coefficient matches this option.
Option (D): cosec A
This option is incorrect because it misses the factor 2. The numerator simplifies to 2(1 + cos A), and after cancelling the common factor (1 + cos A), the expression becomes 2/sin A, not 1/sin A.
Final Answer
The given expression is:
sin A/(1 + cos A) + (1 + cos A)/sin A
After simplification:
= 2/sin A
Therefore:
= 2 cosec A
Correct Option: (B) 2 cosec A
