Verify the Identity (cos(x))/(1-sin(x))=(1+csc(x))/(csc(x))

Problem

cos(x)/(1−sin(x))=(1+csc(x))/csc(x)

Solution

1. Simplify the right side of the equation by using the reciprocal identity csc(x)=1/sin(x)

2. Rewrite the numerator of the right side by finding a common denominator for 1+1/sin(x)

1+1/sin(x)=(sin(x)+1)/sin(x)

3. Divide the fractions on the right side by multiplying the numerator by the reciprocal of the denominator.

(sin(x)+1)/sin(x)/1/sin(x)=(sin(x)+1)/sin(x)⋅sin(x)/1

4. Cancel the common factor sin(x) to simplify the right side to 1+sin(x)

(sin(x)+1)/sin(x)⋅sin(x)/1=1+sin(x)

5. Manipulate the left side by multiplying the numerator and the denominator by the conjugate of the denominator, which is 1+sin(x)

cos(x)/(1−sin(x))⋅(1+sin(x))/(1+sin(x))=(cos(x)*(1+sin(x)))/(1−sin2(x))

6. Apply the Pythagorean identity 1−sin2(x)=cos2(x) to the denominator of the left side.

(cos(x)*(1+sin(x)))/cos2(x)

7. Simplify the fraction by canceling one factor of cos(x) from the numerator and denominator.

(1+sin(x))/cos(x)

8. Re-evaluate the goal as the previous steps show the expressions are not identical as written. Let's re-examine the right side simplification 1+sin(x) and the left side simplification (1+sin(x))/cos(x) There may be a typo in the provided identity, or we can check if cos(x)/(1−sin(x)) equals 1+sin(x) under specific conditions, but algebraically:

cos(x)/(1−sin(x))=(cos(x)*(1+sin(x)))/(1−sin2(x))=(cos(x)*(1+sin(x)))/cos2(x)=(1+sin(x))/cos(x)

9. Simplify the right side again to compare:

(1+csc(x))/csc(x)=1/csc(x)+csc(x)/csc(x)=sin(x)+1

10. Conclude that the left side (1+sin(x))/cos(x) and the right side 1+sin(x) are only equal if cos(x)=1 However, if the task is to verify the identity as given, we show the simplified forms of both sides.

Final Answer

cos(x)/(1−sin(x))≠(1+csc(x))/csc(x)

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