Question 173390
good one.
maybe i got the answer.
you want to prove this equation is true.
here goes:
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let c = csc(x)
let s = sec(x)
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your equation becomes:
{{{c/(1+c) - c/(1-c) = 2s^2}}}
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if you multiply both sides of the equation by (1-c)*(1+c), you get:
{{{c*(1-c) - c*(1+c) = 2s^2*(1-c)*(1+c)}}}
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simplifying, this becomes:
{{{c - c^2 - c + c^2 = 2s^2 * (1-c^2)}}}
which becomes:
{{{-2c^2 = 2s^2 * (1-c^2)}}}
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this is where substitutions come in.
you know that:
{{{sin^2(x) + cos^2(x) = 1}}}
you also know that:
sin(x) = 1/csc(x)
and that:
cos(x) = 1/sec(x)
substituting for sin(x) and cos(x), the equation becomes:
{{{1/csc^2(x) + 1/sec^2(x) = 1}}}
if you multiply both sides of this equation by {{{csc^2(x)*sec^2(x)}}}, the equation becomes:
{{{sec^2(x) + csc^2(x) = sec^2(x)*csc^2(x)}}}
subtract {{{csc^2(x)}}} from both sides of this equation to get:
{{{sec^2(x) = sec^2(x)*csc^2(x) - csc^(x)}}}
simplify this on the right hand side to become:
{{{sec^2(x) = csc^2(x) * (sec^2(x) - 1)}}}
divide both sides of this equation by {{{(sec^2(x) - 1)}}} to get:
{{{sec^2(x)/(sec^2(x) - 1) = csc^2(x)}}}
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this is the same as:
{{{csc^2(x) = sec^2(x)/(sec^2(x) - 1)}}}
and, since we originally let c = csc(x) and we let s = sec(x), this equation now becomes:
{{{c^2 = s^2/(s^2-1)}}}
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we can now substitute for c^2 in the original equation we derived above, which was:
{{{-2c^2 = 2s^2 * (1-c^2)}}}
simplifying this, it becomes:
{{{-2c^2 = 2s^2 - 2s^2*c^2}}}
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substituting for {{{c^2}}}, that equation becomes:
{{{-2*(s^2/(s^2-1)) = 2s^2 - 2s^2*s^2/(s^2-1)}}}
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if we multiply both sides of this equation by {{{s^2-1}}}, we get:
{{{-2s^2 = 2s^2 * (s^2-1) - 2s^2*s^2}}}
this becomes:
{{{-2s^2 = 2s^4 - 2s^2 - 2s^4}}}
which becomes:
{{{-2s^2 = -2s^2}}}
if we multiply both sides of this equation by (-1), it becomes:
{{{2s^2 = 2s^2}}}
which proves that the original equation of:
{{{c/(1+c) - c/(1-c) = 2s^2}}}
is true.
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since we originally substituted c for csc(x), and s for sec(x), the original equation becomes:
{{{csc(x)/(1+csc(x)) - csc(x)/(1-csc(x)) = 2sec^2(x)}}}
and the proven identify becomes:
{{{2sec^2(x) = 2sec^2(x)}}}
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