Question 1161703
If tan²x = 1 + 2tan²y
Show that
cos2x + sin²y = 0
<pre>
tan²x = 1 + 2tan²y

Use the identity

          {{{1+tan^2(theta)=sec^2(theta)}}} 

rewritten as 

          {{{tan^2(theta)=sec^2(theta)-1}}}

{{{sec^2(x)-1 = 1 + 2(sec^2(y)-1)}}}
{{{sec^2(x)-1 = 1+2sec^2(y)-2}}}
{{{sec^2(x)-1 = 2sec^2(y)-1}}}
{{{sec^2(x)=2sec^2(y)}}}
{{{1/cos^2(x)=2(1/cos^2(y))}}}
{{{1/cos^2(x)=2/cos^2(y)}}}
{{{cos^2(y)=2cos^2(x)}}}

Use the identity

          {{{sin^2(theta)+cos^2(theta)=1}}} 

rewritten as 

          {{{cos^2(theta)=1-sin^2(theta)}}}

on the left side

Use the identity 

          {{{cos(2theta)=2cos^2(theta)-1}}}

rewritten as

          {{{2cos^2(theta)=cos(2theta)+1}}}

on the right side

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

Subtract 1 from both sides:

{{{-sin^2(y)=cos(2x)}}}

add sin²(x) to both sides:

{{{0=cos(2x)+sin^2(x)}}}

Swap sides:

{{{cos(2x)+sin^2(x)=0}}} 

Edwin</pre>