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.