Answer:
sec(x)/(tan xsin(x))=cot^2 x+1 = Ture
Step-by-step explanation:
Verify the following identity:
sec(x)/(tan(x) sin(x)) = cot(x)^2 + 1
Hint: | Eliminate the denominator on the left hand side.
Multiply both sides by sin(x) tan(x):
sec(x) = ^?sin(x) tan(x) (cot(x)^2 + 1)
Hint: | Express both sides in terms of sine and cosine.
Write cotangent as cosine/sine, secant as 1/cosine and tangent as sine/cosine:
1/cos(x) = ^?sin(x)/cos(x) sin(x) ((cos(x)/sin(x))^2 + 1)
Hint: | Simplify the right hand side.
((cos(x)/sin(x))^2 + 1) sin(x) (sin(x)/cos(x)) = (((cos(x)^2)/(sin(x)^2) + 1) sin(x)^2)/(cos(x)):
1/cos(x) = ^?(sin(x)^2 (cos(x)^2/sin(x)^2 + 1))/cos(x)
Hint: | Put the fractions in cos(x)^2/sin(x)^2 + 1 over a common denominator.
Put cos(x)^2/sin(x)^2 + 1 over the common denominator sin(x)^2: cos(x)^2/sin(x)^2 + 1 = (cos(x)^2 + sin(x)^2)/sin(x)^2:
1/cos(x) = ^?sin(x)^2/cos(x) (cos(x)^2 + sin(x)^2)/sin(x)^2
Hint: | Cancel down ((cos(x)^2 + sin(x)^2) sin(x)^2)/(sin(x)^2 cos(x)).
Cancel sin(x)^2 from the numerator and denominator. ((cos(x)^2 + sin(x)^2) sin(x)^2)/(sin(x)^2 cos(x)) = (sin(x)^2 (cos(x)^2 + sin(x)^2))/(sin(x)^2 cos(x)) = (cos(x)^2 + sin(x)^2)/cos(x):
1/cos(x) = ^?(cos(x)^2 + sin(x)^2)/cos(x)
Hint: | Eliminate the denominators on both sides.
Multiply both sides by cos(x):
1 = ^?cos(x)^2 + sin(x)^2
Hint: | Use the Pythagorean identity on cos(x)^2 + sin(x)^2.
Substitute cos(x)^2 + sin(x)^2 = 1:
1 = ^?1
Hint: | Come to a conclusion.
The left hand side and right hand side are identical:
Answer: (identity has been verified)
