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(Sec(A) - tan(A))^2 = (1-sin(A))/(1+sin(A))
\n" ); document.write( "The problem with your work so far is that sec^2(A) = 1 + tan^2(A), not 1 - tan^2(A).
\n" ); document.write( "With these problems, it is often helpful to rewrite sec, csc, tan and cot in terms of sin and/or cos. This is especially true for this problem because your RHS is already in terms of just sin(A). So this is what we will do first:
\n" ); document.write( "(1/cos(A) - sin(A)/cos(A))^2 = (1-sin(A))/(1+sin(A))
\n" ); document.write( "On the LHS, the two fractions have the same denominator so they can be combined. And since the RHS is just one term, this looks like a good idea:
\n" ); document.write( "((1-sin(A))/cos(A))^2 = (1-sin(A))/(1+sin(A))
\n" ); document.write( "We can square the fraction on the LHS:
\n" ); document.write( "((1-sin(A))^2/cos^2(A) = (1-sin(A))/(1+sin(A))
\n" ); document.write( "This looks promising. We already have 1-sin(A) as a factor of the numerator on both sides of the equation. We don't want any cos but, since it's squared, this is easy to turn into an expression of sin(A):
\n" ); document.write( "((1-sin(A))^2/(1-sin^2(A)) = (1-sin(A))/(1+sin(A))
\n" ); document.write( "What we want is
- One of the (1-sin(A)) factors in the numerator on the LHS to \"disappear\".
- The 1-sin^2(A) in the denominator on the LHS to turn into 1+sin(A) somehow.
\n" ); document.write( "As soon as we recognize that 1-sin^2(A) is a difference of squares and that it will factor into (1+sin(A))(1-sin(A)), we will realize that this will take care of both of our goals:
\n" ); document.write( "((1-sin(A))^2/((1+sin(A))(1-sin(A)) = (1-sin(A))/(1+sin(A))
\n" ); document.write( "On the LHS, (1-sin(A)) is a factor of both the numerator and denominator so they will cancel:
\n" ); document.write( "(1-sin(A)/(1+sin(A)) = (1-sin(A))/(1+sin(A))
\n" ); document.write( "and we're done. \n" ); document.write( "