How do you simplify ${tan}^{2} x ({csc}^{2} x - 1)$ $tan^2x(csc^2x-1)$ ?

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How do you simplify ${tan}^{2} x ({csc}^{2} x - 1)$ $tan^2x(csc^2x-1)$ ?

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Answer 1 · 2015-07-26T08:10:03

By using the Trigonometric Identity : ${sin}^{2} x + {cos}^{2} x = 1$ $sin^2x+cos^2x=1$

Explanation:

Divide both sides of the above identity by ${sin}^{2} x$ $sin^2x$ to obtain,

$\frac{{sin}^{2} x}{{sin}^{2} x} + \frac{{cos}^{2} x}{{sin}^{2} x} = \frac{1}{{sin}^{2} x}$ $sin^2x/(sin^2x)+cos^2x/sin^2x=1/sin^2x$

$\Rightarrow 1 + \frac{1}{\frac{{sin}^{2} x}{{cos}^{2} x}} = {csc}^{2} x$ $=>1+1/(sin^2x/cos^2x)=csc^2x$

$\Rightarrow 1 + \frac{1}{{tan}^{2} x} = {csc}^{2} x$ $=>1+1/tan^2x=csc^2x$

$\Rightarrow {csc}^{2} x - 1 = \frac{1}{{tan}^{2} x}$ $=>csc^2x-1=1/tan^2x$

Now, we are able to write : ${tan}^{2} x ({csc}^{2} x - 1)$ $tan^2x(csc^2x-1)" "$ as ${tan}^{2} x (\frac{1}{{tan}^{2} x})$ $" "tan^2x(1/tan^2x)$

and the result is $1$ $color(blue)1$

Answer 2 · 2015-07-27T04:22:48

Simplify: ${tan}^{2} x ({csc}^{2} x - 1)$ $tan^2 x(csc^2 x - 1)$

Explanation:

$\frac{{sin}^{2} x}{{cos}^{2} x} (\frac{1}{{sin}^{2} x} - 1) = (\frac{{sin}^{2} x}{{cos}^{2} x}) (\frac{1 - {sin}^{2} x}{{sin}^{2} x}) =$ $sin^2 x/cos^2x(1/sin^2 x - 1) = (sin^2 x/cos^2 x)((1 - sin^2 x)/sin^2 x) =$

$= \frac{{sin}^{2} x}{{cos}^{2} x} (\frac{{cos}^{2} x}{{sin}^{2} x})$ $= sin^2 x/cos^2 x(cos^2 x /sin^2 x)$ = 1.

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How do you simplify ${tan}^{2} x ({csc}^{2} x - 1)$ $tan^2x(csc^2x-1)$ ?

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How do you simplify ${tan}^{2} x ({csc}^{2} x - 1)$ $tan^2x(csc^2x-1)$ ?