#1 cscθ-cotθ
#2 cosθ(tanθ-secθ)
#3 tanθ+sinθ/cscθ+cotθ
nice expressions
what do you want to do with them?
the answer an how to do it
I don't see any questions. How do I provide answers?
For instance, cscθ-cotθ can be rewritten in many ways, using various combinations of sin,cos, etc. Just plopping a few expressions on the net doesn't elicit any responses. Are these problems in some section of your text? If so, there is usually an instruction on what you are to do with the expressions.
To simplify the given expressions, we can use trigonometric identities. Let's break down each of the expressions and simplify them step by step:
#1 cscθ - cotθ:
Step 1: Recall the definitions of cscθ and cotθ:
- cscθ = 1/sinθ
- cotθ = 1/tanθ = cosθ/sinθ
Step 2: Substitute the definitions into the expression:
cscθ - cotθ = 1/sinθ - cosθ/sinθ
Step 3: Combine the fractions with the same denominator:
(1 - cosθ)/sinθ
#2 cosθ(tanθ - secθ):
Step 1: Recall the definitions of tanθ and secθ:
- tanθ = sinθ/cosθ
- secθ = 1/cosθ
Step 2: Substitute the definitions into the expression:
cosθ(tanθ - secθ) = cosθ(sinθ/cosθ - 1/cosθ)
Step 3: Simplify the expression:
cosθ(sinθ - 1)/cosθ = sinθ - 1
#3 tanθ + sinθ/cscθ + cotθ:
Step 1: Recall the definitions of cscθ and cotθ:
- cscθ = 1/sinθ
- cotθ = 1/tanθ = cosθ/sinθ
Step 2: Substitute the definitions into the expression:
tanθ + sinθ/(1/sinθ) + (cosθ/sinθ)
Step 3: Simplify the expression:
tanθ + sinθ * sinθ + cosθ/sinθ = tanθ + sin²θ + cosθ/sinθ
These are the simplified forms of the given expressions using trigonometric identities.