Evaluate the following expression.(x ≠ πn, y ≠ π/12 + πn, where n is a whole number)
A)7 tan 6x + 7 cot 6y
B)7 tan 6y - 7 cot 6x
C)7 tan 6y + 7 cot 6x
D)- (7 tan 6y + 7 cot 6x)
E)7 tan 6x - 7 cot 6y

Question

Evaluate the following expression.(x ≠ πn, y ≠ π/12 + πn, where n is a whole number) ​  ​
A)7 tan 6x + 7 cot 6y 
B)7 tan 6y - 7 cot 6x 
C)7 tan 6y + 7 cot 6x 
D)- (7 tan 6y + 7 cot 6x) 
E)7 tan 6x - 7 cot 6y

Quizplus · Accepted Answer

We can use the identity:

tan(a+b) = (tan(a) + tan(b))/(1 - tan(a)tan(b))

cot(a+b) = (cot(a)cot(b) - 1)/(cot(a) + cot(b))

to write:

7 tan(6x + 6y) = 7[(tan(6x) + tan(6y))/(1 - tan(6x)tan(6y))]

7 cot(6x + 6y) = 7[(cot(6x)cot(6y) - 1)/(cot(6x) + cot(6y))]

Adding these two expressions, we get:

7 tan(6x + 6y) + 7 cot(6x + 6y) = 7[(tan(6x) + tan(6y))/(1 - tan(6x)tan(6y))] + 7[(cot(6x)cot(6y) - 1)/(cot(6x) + cot(6y))]

We can simplify this expression using the following identity:

cot(x) = 1/tan(x)

which gives:

7[(tan(6x) + tan(6y))/(1 - tan(6x)tan(6y))] + 7[(cot(6x)cot(6y) - 1)/(cot(6x) + cot(6y))]

= 7[(tan(6x) + tan(6y))/(1 - tan(6x)tan(6y))] + 7[(1/tan(6x) * 1/tan(6y) - 1)/(1/tan(6x) + 1/tan(6y))]

= 7[(tan(6x) + tan(6y))/(1 - tan(6x)tan(6y))] + 7[(1/(tan(6x)tan(6y)) - (tan(6x) + tan(6y))/(tan(6x)tan(6y)))/(tan(6x) + tan(6y))/(tan(6x)tan(6y))]

= 7[(tan(6x) + tan(6y))/(1 - tan(6x)tan(6y))] - 7[(tan(6x) + tan(6y))/(tan(6x)tan(6y) - 1)]

= 7[(tan(6x) + tan(6y))(tan(6x)tan(6y) - 1) - (1 - tan(6x)tan(6y))(tan(6x) + tan(6y))]/(1 - tan(6x)tan(6y))(tan(6x)tan(6y) - 1)

= 7[(tan(6x)tan(6x)tan(6y) - tan(6y))(tan(6x)tan(6y) - 1) + (tan(6x) - tan(6x)tan(6y) - tan(6y) + 1)]/(1 - tan(6x)tan(6y))(tan(6x)tan(6y) - 1)

= 7[(tan(6x)tan(6x)tan(6y)tan(6y) - tan(6x)tan(6y) - tan(6x)tan(6y) + tan(6y))(tan(6x)tan(6y) - 1) + (tan(6x) + 1 - tan(6x)tan(6y) - tan(6y))]/(1 - tan(6x)tan(6y))(tan(6x)tan(6y) - 1)

= 7[(tan(6x)tan(6y) - tan(6x)tan(6y) - tan(6y)tan(6x) + tan(6y) + tan(6x) - tan(6x)tan(6y) - tan(6y) + 1)]/(1 - tan(6x)tan(6y))(tan(6x)tan(6y) - 1)

= 7(1 - tan(6y)tan(6x))/(1 - tan(6y)tan(6x)) = 7

Therefore, the expression simplifies to 7, which is choice C.

Evaluate the Following Expression