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Home Q & A Board Science-Related Homework Help Mathematics
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datewing datewing
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11 years ago
Having a little trouble with the use of trigonometric identities in proving some of these statements. These are what i need to prove


sin^2 x (csc^2 x + sec^2x) = sec^2x

cos 0 (sin 0 +(cos^2 0)/sin 0) = cot 0

the zeros on the last problem take the place of the little 0 with the line through the middle that you see in math and that i didnt know how to replicate on the computer. Thanks to anyone who can help.
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wrote...
#1
11 years ago
With these, always start by converting everything into sin and cos - that way you can see what you're working with  (and you only have to learn the identities for sin and cos, and not for tan, csc, sec or cot)...


LHS = Sin^2x * (1/Sin^2x + 1/cos^2x)
= Sin^2x (cos^2x + Sin^2x)/[Sin^2xCos^2x]

remember that Sin^2x + cos^2x = 1
= sin^2x/[SIn^2xCos^2x]
Cancel out the SIn^2x
= 1/cos^2x = sec^2 x = RHS

RHS = cot0 = 1/tan0 = 1/(sin0/cos0) = cos0/sin0
LHS = cos 0 (sin 0 +(cos^2 0)/sin 0)
= cos0/sin0) * (sin^20 + cos^20)
= cos0/sin0 * 1
= RHS
This doesn't equal the right hand side unless you meant....
wrote...
#2
11 years ago
sin^2 x (csc^2 x + sec^2x) = sec^2x
LHS = sin^2 x (csc^2 x + sec^2x)
       = sin^2 x csc^2x + sin^2x sec^2x
       = 1 + (1-cos^2x)sec^2x                       (sinx cscx =1;   sin^2x = 1-cos^2x)
       = 1 + sec^2x - 1                                 (secx cosx=1)
       = sec^2x  RHS , proved

cos 0 (sin 0 +(cos^2 0)/sin 0) = cot 0    

LHS = cos 0 (sin 0 +(cos^2 0)/sin 0)
        = cos 0 ( sin^2 0 + cos^2 0 )/sin 0
        = cos 0/sin 0    [sin^2 0 + cos^2 0=1]
        = cot 0 =RHS
wrote...
#3
11 years ago
sin^2(x) * (csc^2(x) + sec^2(x)) = sec^2(x)
sin^2(x) * (1/sin^2(x) + 1/cos^2(x)) = sec^2(x)
1 + sin^2(x)/cos^2(x) = sec^2(x)
(sin^2(x) + cos^2(x)) / cos^2(x) = sec^2(x)
1 / cos^2(x) = sec^2(x)
sec^2(x) = sec^2(x)

cos(x) * (sin(x) + cos^2(x))/sin(x) = cot(x)
cos(x) * ((sin^2(x) + cos^2(x))/sin(x) = cot(x)
cos(x) / sin(x) = cot(x)
cot(x) = cot(x)
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RiveraCIRiveraCI
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#4
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11 years ago
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wrote...
#5
11 years ago
LHS                                                                  RHS
sin^2x(1/sin^2x + 1/cos^2x)                               sec^2x
sin^2x(cos^2x + sin^2x)/cos^2x sin^2x
where cos^2x + sin^2x=1
then
sin^2x(1/cos^2x sin^2x)
sin^2x/(cos^2x sin^2x)
1/cos^2x
sec^2x =                                                       sec^2x


cos 0 (sin 0 +(cos^2 0)/sin 0)
cos 0 (sin^2 0 + cos^2 0)/sin 0
where sin^2 0 + cos^2 0 =1
then
cos 0 (1)sin 0
cos 0/sin 0
cot 0                                  =              cot 0

hope i have answered your question.good luck.
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