document.write( "Question 349080: Simplify: 1 - csc^2 (theta)/cot^2 (theta) \n" ); document.write( "

Algebra.Com's Answer #249594 by jsmallt9(3758) $\"\"$ $\"About$

You can put this solution on YOUR website!
(Algebra.com's software will not do theta. So I will use \"x\" instead.)
\n" ); document.write( " $\"%281+-+csc%5E2%28x%29%29%2Fcot%5E2%28x%29\"$

\n" ); document.write( "When trying to manipulate and simplify expressions with trig functions, it is often helpful to rewrite the expression using only sin and/or cos. The other 4 functions can be written in terms of sin/cos. Using this idea on your expression, and using the facts that csc = 1/sin and cot = cos/sin, we get:
\n" ); document.write( " $\"%281+-+%281%2Fsin%5E2%28x%29%29%29%2F%28cos%5E2%28x%29%2Fsin%5E2%28x%29%29\"$
\n" ); document.write( "With $\"sin%5E2%28x%29\"$ as the denominator of both \"little\" fractions, we can simplify this expression by multiplying the numerator and denominator of the \"big\" fraction by $\"sin%5E2%28x%29\"$ :
\n" ); document.write( "

\n" ); document.write( "On top we will need to use the Distributive Property to multiply:
\n" ); document.write( " $\"%28sin%5E2%28x%29+-+1%29%2Fcos%5E2%28x%29\"$
\n" ); document.write( "We know that $\"cos%5E2%28x%29+=+1+-+sin%5E2%28x%29\"$ . So what is $\"sin%5E2%28x%29+-+1\"$ ? Answer: $\"-cos%5E2%28x%29\"$ !! Substituting this into the expression we get:
\n" ); document.write( " $\"%28-cos%5E2%28x%29%29%2Fcos%5E2%28x%29\"$
\n" ); document.write( "The cosines cancel leaving
\n" ); document.write( " $\"%28-1%29%2F1+=+-1\"$
\n" ); document.write( "Your expression simplifies down to -1! \n" ); document.write( "

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