SOLUTION: Verify the identity.
a)(csc(-t)-sin(-t))/(sin(-t))=cot^(2)t
b)ln cotx= -ln tanx
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Question 148202: Verify the identity.
a)(csc(-t)-sin(-t))/(sin(-t))=cot^(2)t
b)ln cotx= -ln tanx
Found 2 solutions by jim_thompson5910, mangopeeler07:
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
a)
Note: I'm only algebraically manipulating the left side. I'm showing the right side for comparison.
(csc(-t)-sin(-t))/(sin(-t))=cot^(2)t ... Start with the given equation
(1/sin(-t)-sin(-t))/(sin(-t))=cot^(2)t .... Replace csc(-t) with 1/sin(-t)
-(-1/sin(t)+sin(t))/(sin(t))=cot^(2)t .... Replace each sin(-t) with -sin(t)
(1/sin(t)-sin(t))/(sin(t))=cot^(2)t .... Simplify
(1/sin(t)-sin^2(t)/sin(t))/(sin(t))=cot^(2)t ... Rewrite sin(t) as sin^2(t)/sin(t)
((1-sin^2(t))/sin(t))/(sin(t))=cot^(2)t ... Combine the fractions in the numerator
((1-sin^2(t))/sin^2(t)=cot^(2)t ... Divide the fractions
(cos^2(t))/sin^2(t)=cot^(2)t ... Replace 1-sin^2(t) with cos^2(t)
cot^2(t)=cot^(2)t .... Replace (cos^2(t))/sin^2(t) with cot^2(t)
So this verifies the identity.
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b)
Note: I'm only algebraically manipulating the left side. I'm showing the right side for comparison.
ln(cot(x))= -ln(tan(x)) ... Start with the given equation.
ln(1/tan(x))= -ln(tan(x)) ... Rewrite cot(x) as 1/tan(x)
ln((tan(x))^(-1))= -ln(tan(x)) ... Rewrite 1/tan(x) as (tan(x))^(-1)
-1*ln(tan(x))= -ln(tan(x)) ... Rewrite the left side using the identity
-ln(tan(x))= -ln(tan(x)) .... Multiply
So this verifies the identity.
Answer by mangopeeler07(462) (Show Source): You can put this solution on YOUR website!
a)(csc(-t)-sin(-t))/(sin(-t))=cot^(2)t
Kick out those t's to make this thing look better.
(csc-sin)/(sin)=cot^2
Plug in identities you already know.
cot^2=cos^2/sin^2
csc=1/sin
(1/sin)-sin/sin)=cos^2/sin^2
Do 1/sin-sin.
(1/sin)-sin=(1/sin)-(sin^2/sin) or (1-sin^2)/sin
1-sin^2=cos^2
So (1-sin^2)/sin = cos^2/sin.
Plug that back into ((1/sin)-sin)/sin=cos^2/sin^2
and get (cos^2/sin)/sin=cos^2/sin^2
Change (cos^2/sin)/sin to (cos^2/sin)(1/sin) and get cos^2/sin^2, or cot^2.
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