Prove the identities tan ? + cot ? = csc ?/cos ?

Question

So lost 

Answer 1

This is a case where the best approach might be to convert both sides to some expression based on sin and cos, to see if they come out the same. 

 

Let's look at each side:

 

LEFT SIDE

tan x + cot x

= tan x + 1/tan x                         (since cot x is the reciprocal of tan x)

= (tan² x + 1) / tan x                   (using tan x as a common denominator to add the two terms)

= sec² x / tan x                            (by one of the pythagorean identities)

= (1 / cos² x) / (sin x / cos x)       (since sec x = 1/cos x, and since tan x = sin x / cos x)

= (1 / cos² x) * (cos x / sin x)      (since dividing by a fraction is the same as multiplying by its reciprocal)

= cos x / (sin x cos² x)                 (multiplying the two fractions)

= 1 / (sin x cos x)                         (canceling out cos x from the numerator and denominator)

 

RIGHT SIDE

csc x / cos x

= (1 / sin x) / cox x                      (since csc x is the reciprocal of sin x)

= 1 / (sin x cox x)                         (by the rules of division)

 

So both sides end up equal to 1 / (sin x cox x), which proves that they are equal to each other.