verify the following identity:
tanx+cotx/cscx=secx
looks like we have some "homework dumping" going on here.
I will do one more, and let you looks at my methods.
Then you tell me where you are having problems
Again, you must mean
(tanx + cotx)/cscx = secx
LS = (sinx/cosx + cosx/sinx)(sinx)
= (sin^2 x + cos^2 x)/(sinxcosx) (sinx)
= 1/cosx
= sec x
= RS
To verify the given trigonometric identity:
We can start with the left side of the equation:
tan(x) + cot(x) / csc(x)
Now, let's rewrite cot(x) and csc(x) in terms of sine and cosine:
cot(x) = 1/tan(x) (Reciprocal identity: cot(x) = 1/tan(x))
csc(x) = 1/sin(x) (Reciprocal identity: csc(x) = 1/sin(x))
Substituting these values, we have:
tan(x) + 1/tan(x) / 1/sin(x)
Simplifying further:
tan(x) + sin(x)/cos(x)
Now, let's find the common denominator, which is cos(x):
(tan(x)*cos(x) + sin(x)) / cos(x)
Now, let's simplify the numerator:
(sin(x)/cos(x) * cos(x) + sin(x)) / cos(x)
sin(x) + sin(x) / cos(x)
Now, let's combine the similar terms:
2sin(x) / cos(x)
We can rewrite 2sin(x) as sin(2x):
sin(2x) / cos(x)
Now, let's use the double-angle identity for sine:
sin(2x) = 2sin(x)cos(x)
Substituting this value, we get:
2sin(x)cos(x) / cos(x)
Canceling out the common factor of cos(x) in the numerator and denominator:
2sin(x)
Finally, we can use the reciprocal identity of sine:
2sin(x) = 2/csc(x)
So, the simplified expression is:
2/csc(x)
Now, comparing this with the right side of the original equation, which is sec(x), we find that they are equal.
Therefore, the left side and the right side of the equation are indeed equal, verifying the given trigonometric identity:
tan(x) + cot(x) / csc(x) = sec(x)