Question 1106380
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i) tanē(x) - sinē(x) = tanē(x) sinē(x) 

change tanē(x) to sinē(x)/cosē(x)

{{{sin^2(x)/cos^2(x)-sin^2(x)}}}

Get LCD of cosē(x)

{{{(sin^2(x)-sin^2(x)cos^2(x))/cos^2(x)}}}

Factor out sinē(x) on top:

{{{(sin^2(x)(1-cos^2(x)))/cos^2(x)}}}

Replace 1-cosē(x) by sinē(x)

{{{(sin^2(x)(sin^2(x)))/cos^2(x)}}}

Break into a product:

{{{(sin^2(x)^"")*((sin^2(x))/cos^2(x))}}}

Replace {{{(sin^2(x))/cos^2(x)}}} by {{{tan^2(x)}}}

{{{sin^2(x)^""*tan^2(x)}}}

Turn backward

{{{tan^2(x)^""*sin^2(x)}}}

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ii) {{{(1+ cos(x) + cos(2x))/ (sin(x) + sin(2x))}}}{{{""=""}}}{{{cot(x)}}}

Use identities for cos(2x) and sin(2x)

{{{(1+ cos(x) + 2cos^2(x)-1)/ (sin(x) + 2sin(x)cos(x))}}}

{{{(cos(x) + 2cos^2(x))/ (sin(x) + 2sin(x)cos(x))}}}

Factor out common factors on top and bottom:

{{{(cos(x)(1^"" + 2cos(x)))/ (sin(x)(1^"" + 2cos(x)))}}}

{{{(cos(x)(cross(1 + 2cos(x))))/ (sin(x)(cross(1 + 2cos(x))))}}}

{{{cos(x)/sin(x)}}}

{{{cot(x)}}}

Edwin</pre>