Verify the Identity:
csc(x)+sec(x)/sin(x)+cos(x)=cot(x)+tan(x)
the left side of the equation is all one term.
Verify:
(csc(x)+sec(x))/(sin(x)+cos(x))=cot(x)+tan(x)
Left hand side
(csc(x)+sec(x))/(sin(x)+cos(x))
=(1/sin(x)+1/(cos(x))/(sin(x)+cos(x))
=((cos(x)+sin(x))/(sin(x)cos(x))/(sin(x)+cos(x))
=1/(sin(x)cos(x))
Right hand side:
cot(x)+tan(x)
=cos(x)/sin(x) + sin(x)/cos(x)
=(cos²(x) + sin²(x))/(sin(x)+cos(x))
=1/(sin(x)+cos(x))
So the identity is verified.
To verify the given identity, we should manipulate the left side of the equation to obtain a single term.
Let's start by simplifying each trigonometric expression on the left side of the equation:
1. Using the reciprocal identity csc(x) = 1/sin(x):
csc(x) + sec(x) = 1/sin(x) + 1/cos(x)
2. Finding a common denominator:
csc(x) + sec(x) = (cos(x) + sin(x))/(sin(x) * cos(x))
3. Applying the distributive property to the denominator:
csc(x) + sec(x) = (cos(x) + sin(x))/(sin(x) * cos(x))
4. Using the reciprocal identity sec(x) = 1/cos(x):
csc(x) + sec(x) = (cos(x) + sin(x))/(sin(x) * cos(x))
Now let's simplify the right side of the equation:
1. Using the reciprocal identity cot(x) = 1/tan(x):
cot(x) + tan(x) = 1/tan(x) + tan(x)
2. Finding a common denominator:
cot(x) + tan(x) = (1 + tan^2(x))/(tan(x))
3. Using the Pythagorean identity: 1 + tan^2(x) = sec^2(x):
cot(x) + tan(x) = (sec^2(x))/(tan(x))
Now that we have simplified both sides of the equation, we can compare them:
Left side: (cos(x) + sin(x))/(sin(x) * cos(x))
Right side: (sec^2(x))/(tan(x))
As you can see, the left side and the right side are not identical. Therefore, the given equation csc(x) + sec(x)/sin(x) + cos(x) = cot(x) + tan(x) is not an identity.