use the quotient and reciprocal identities to simplify the given expression
cot t sin t
csc t sin t
tan t cot t
cot t sec t
cot t sin t
csc t sin t
tan t cot t
cot t sec t
To simplify the given expressions using quotient and reciprocal identities, we need to rewrite the trigonometric functions in terms of sine and cosine functions.
1. Simplifying cot t / sin t:
We know that cot t is equal to cos t / sin t.
So, cot t / sin t = (cos t / sin t) / sin t.
Using the quotient identity for sine, we can simplify further:
cot t / sin t = cos t / (sin t * sin t).
This is the simplified form of cot t / sin t.
2. Simplifying csc t / sin t:
We know that csc t is equal to 1 / sin t.
So, csc t / sin t = (1 / sin t) / sin t.
Using the quotient identity for sine, we can simplify further:
csc t / sin t = 1 / (sin t * sin t).
This is the simplified form of csc t / sin t.
3. Simplifying tan t / cot t:
We know that tan t is equal to sin t / cos t.
So, tan t / cot t = (sin t / cos t) / (cos t / sin t).
Using the quotient identity for cosine, we can simplify further:
tan t / cot t = (sin t / cos t) * (sin t / cos t).
This is the simplified form of tan t / cot t.
4. Simplifying cot t / sec t:
We know that sec t is equal to 1 / cos t.
So, cot t / sec t = (cos t / sin t) / (1 / cos t).
Using the quotient identity for cosine, we can simplify further:
cot t / sec t = (cos t / sin t) * (cos t / 1).
This is the simplified form of cot t / sec t.
Remember, it is important to understand the basic trigonometric identities and properties to simplify trigonometric expressions.