Expert: Sherman D. - 4/19/2009

Question
"all x's stand for theta

(1+sinx)(1-sinx)(secx+1)(secx+1)

[sinx-(sinx)(cosx)]/1-cosx

2sinx/1-cos^2(x)

[sin^2(x)/cos^2x] +sinx times cscx=sec^2(x)

[cosx/sinxcotx]/1

1-secxcosx=tanxcotx-1

[1/tanx-secx]  + [1/tanx+secx]=-2tanx

cosx ( tan^2x+10)=secx

sec^2xcotx-cotx=tanx

sin^3x+sinxcos^2x=sinx

cot^2x-cos^2x=cot^2x times cos^2x

Answer
for theta, you type it like this sin(A), so if by theses you mean

(1 + sin(A))(1 - sin(A))(sec(A) + 1)(sec(A) + 1)
(1 - sin(A)^2)(sec(A)^2 + 2sec(A) + 1)
(cos(A)^2)((1/cos(A))^2 + (2/cos(A)) + 1)
(cos(A)^2/cos(A)^2) + (2cos(A)^2/cos(A)) + cos(A)^2
1 + 2cos(A) + cos(A)^2
ANS : cos(A)^2 + 2cos(A) + 1

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[sin(A) - (sin(A))(cos(A))]/(1 - cos(A))
(sin(A)(1 - cos(A)))/(1 - cos(A))
the (1 - cos(A))s cancel
ANS : sin(A)

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(2sin(A))/(1 - cos(A)^2)
(2sin(A))/(sin(A)^2)
ANS : 2/sin(A) or 2csc(A)

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[sin(A)^2/cos(A)^2] + sin(A)csc(A) = sec^2(A)
tan(A)^2 + sin(A)(1/sin(A)) = sec(A)^2
tan(A)^2 + 1 = sec(A)^2
sec(A)^2 = sec(A)^2

i could do this the long way if you like, but trust me, it will end up the same way.

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Are you sure you don't mean
[cos(A)/(sin(A)cot(A))] = 1
(cos(A)/(sin(A)(cos(A)/sin(A)))) = 1
(cos(A)/(sin(A)cos(A)/sin(A))) = 1
cos(A)/cos(A) = 1
1 = 1

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1 - sec(A)cos(A) = tan(A)cot(A) - 1
1 - (1/cos(A))cos(A) = tan(A)(1/tan(A)) - 1
1 - 1 = 1 - 1
0 = 0

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[1/(tan(A) - sec(A))]  + [1/(tan(A) + sec(A))] = -2tan(A)
Multiply the left side by (tan(A) - sec(A))(tan(A) + sec(A))
(tan(A) + sec(A) + tan(A) - sec(A))/(tan(A)^2 - sec(A)^2) = -2tan(A)
(2tan(A))/((sin(A)/cos(A))^2 - (1/cos(A))^2)) = -2tan(A)
(2tan(A))/((sin(A)^2 - 1)/cos(A)^2) = -2tan(A)
(2tan(A))/((-1 + sin(A)^2)/cos(A)^2) = -2tan(A)
(2tan(A))/(-(1 - sin(A)^2)/cos(A)^2) = -2tan(A)
(-2tan(A))/(cos(A)^2/cos(A)^2) = -2tan(A)
-2tan(A) = -2tan(A)

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Did you mean
cos(A)(tan(A)^2 + 1) = sec(A)
cos(A)sec(A)^2 = sec(A)
cos(A)(1/cos(A)^2) = sec(A)
1/cos(A) = sec(A)
sec(A) = sec(A)

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sec(A)^2cot(A) - cot(A) = tan(A)
cot(A)(sec(A)^2 - 1) = tan(A)
(1/tan(A))tan(A)^2 = tan(A)
tan(A)^2/tan(A) = tan(A)
tan(A) = tan(A)

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sin(A)^3 + sin(A)cos(A)^2 = sin(A)
sin(A)(sin(A)^2 + cos(A)^2) = sin(A)
sin(A) = sin(A)

if your wondering, its because sin(A)^2 + cos(A)^2 = 1

if you won't, you can thank of it like this

y^2 + x^2 = 1
or
(y/r)^2 + (x/r)^2 = 1
(y^2 + x^2)/(r^2) = 1
x^2 + y^2 = r^2

which is the equation for a circle.

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cot(A)^2 - cos(A)^2 = cot(A)^2cos(A)^2
(cos(A)^2/sin(A)^2) - cos(A)^2 = (cos(A)^2/sin(A)^2)(cos(A)^2)
(cos(A)^2 - sin(A)^2cos(A)^2)/sin(A)^2 = (cos(A)^4/sin(A)^2)
(cos(A)^2(1 - sin(A)^2)))/sin(A)^2 = (cos(A)^4/sin(A)^2)
(cos(A)^2 * cos(A)^2)/sin(A)^2 = (cos(A)^4/sin(A)^2)
cos(A)^4/sin(A)^2 = cos(A)^4/sin(A)^2

i did that one the long way to show you why it's equal.

more info found at http://math2.org/math/trig/identities.htm