Expert: Sherman D. - 11/28/2006

Question
1. Reduce (csc^2 x - sec^2 X) to an expression containing only tan x.

2.Verify the following identities.
a.(sin B+cos B) (sin B-cos B) = 2 sin^2 B-1
b.(1-cos^2 y+sin^2 y)^2+4 sin^2 y cos^2 y = 4 sin^2 y
c.tan^2 0 sec^2 0-sec^2 0+1 = tan^4 0
d.sin 0/csc 0 + cos 0/sec 0 = 1
e. sin x tan^2 x cot^3 x = cos x
f.sec^2 X/sec^2 X-1 =csc^2 X

Answer
Lucky for you i love trying to figure these out.

1.) Reduce (csc^2 x - sec^2 X) to an expression containing only tan x.

(csc(x))^2 - (sec(x))^2

csc(x)^2 = cot(x)^2 + 1
sec(x)^2 = tan(x)^2 + 1

cot(x)^2 + 1 - (tan(x)^2 + 1)
cot(x)^2 + 1 - tan(x)^2 - 1
cot(x)^2 - tan(x)^2
(1/tan(x))^2 - tan(x)^2
(1 - tan(x)^4)/(tan(x)^2)

ANS : (1 - tan(x)^4)/(tan(x)^2) or tan(x)^(-2) - tan(x)^2

So unless you typed something wrong, thats what i got. Since you wanted only tan(x), i didn't know for sure how to put it.

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2.) Verify the following identities.

a.)
(sin(B) + cos(B))*(sin(B) - cos(B)) = 2(sin(B))^2 - 1

sin(B)^2 - sin(B)cos(B) + sin(B)cos(B) - cos(B)^2 = 2sin(B)^2 - 1

sin(B)^2 - cos(B)^2 = 2sin(B)^2 - 1

cos(B)^2 = 1 - sin(B)^2

sin(B)^2 - (1 - sin(B)^2) = 2sin(B)^2 - 1
sin(B)^2 - 1 + sin(B)^2 = 2sin(B)^2 - 1
2sin(B)^2 - 1 = 2sin(B)^2 - 1

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b.)
(1 - (cos(y))^2 + (sin(y))^2)^2 + 4(sin(y))^2(cos(y))^2  = 4(sin(y))^2

1 - cos(y)^2 = sin(y)^2

(sin(y)^2 + sin(y)^2)^2 + 4(sin(y))^2(1 - sin(y)^2) = 4sin(y)^2

(2sin(y)^2)^2 + 4sin(y)^2 - 4sin(y)^4 = 4sin(y)^2
4sin(y)^4 + 4sin(y)^2 - 4sin(y)^4 = 4sin(y)^2
4sin(y)^2 = 4sin(y)^2


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I will use A instead of 0, just because its easier to read.
The true symbol is È.

c.)
(tan(A))^2(sec(A))^2 - (sec(A))^2 + 1 = (tan(A))^4
(tan(A)^2 * sec(A))^2 - (sec(A)^2 - 1) = tan(A)^4
(tan(A)^2 * sec(A))^2 - tan(A)^2 = tan(A)^4
(tan(A)^2)(sec(A)^2 - 1) = tan(A)^4
(tan(A)^2)(tan(A)^2) = tan(A)^4
tan(A)^4 = tan(A)^4

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d.)
((sin(A))/(csc(A))) + ((cos(A))/(sec(A))) = 1
(sin(A) / (1/sin(A))) + (cos(A) / (1/cos(A))) = 1
((sinA/1)/(1/sinA)) + ((cosA/1)/(1/cosA)) = 1
sinA^2 + cosA^2 = 1
1 = 1

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e.)
sin(x) * (tan(x))^2 * (cot(x))^3 = cos(x)
sin(x) * tan(x)^2 * cot(x)^2 * cot(x) = cos(x)
sin(x) * (tan(x)^2) * (1/(tan(x)))^2 * cot(x) = cos(x)
sin(x) * cot(x) = cos(x)
sin(x) * (cos(x)/sin(x)) = cos(x)
(sin(x)cos(x))/sin(x) = cos(x)
cos(x) = cos(x)

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f.)
(((sec(x))^2)/((sec(x))^2 - 1) = (csc(x))^2

sec(x)^2 - 1 = tan(x)^2

(sec(x)^2)/(tan(x)^2) = (csc(x))^2

sec(x)^2 = (1/cos(x))^2

(1/cos(x)^2) / (tan(x)^2) = (csc(x))^2
(1/cos(x)^2) / (sin(x)/cos(x))^2 = (csc(x))^2
((1/cos(x)) / (sin(x)/cos(x))^2 = (csc(x))^2
((1/cos(x)) * (cos(x)/sin(x))^2 = (csc(x))^2
(cos(x)/(sin(x)cos(x)))^2 = (csc(x))^2
(1/sin(x))^2 = (csc(x))^2
(csc(x))^2 = (csc(x))^2

More Info found at www.math.com/tables/trig/identities.htm

and if you have anymore questions, try typing it like i have so that i can understand you more.