Using algebra to prove identities

Question

Use algebra to prove the identity: cos(x)/1-sin(x) − tan(x) = 1 cos(x)

Answer 1

Hi Erin, should the right side of the equation be 1 / cos(x) instead of 1 cos(x)? If it is, then the solution is:

 

cos(x) / (1 - sin(x)) - tan(x) = 1 / cos(x)

 

cos(x) / (1 - sin(x)) - sin(x) / cos(x) = 1 / cos(x)

 

The common denominator on the left side of the equation is (1 - sin(x)) * cos(x).

 

Multiply the first term by cos(x) and the second term by (1 - sin(x)) and place over the common denominator:

 

(cos(x) * cos(x) - sin(x) * (1 - sin(x))) / ((1 - sin(x)) * cos(x)) = 1 / cos(x)

 

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

 

Cos²(x) + sin²(x) = 1, then:

 

(1 - sin(x)) / ((1 - sin(x)) * cos(x)) = 1 / cos(x)

 

the (1 - sin(x)) cancels out leaving:

 

1 / cos(x) = 1 / cos(x)

 

Questions?