Question 65913
<b><font size = 8>Answer by Edwin McCravy(279)
<pre><font size = 4><b>
Prove that this equation is an identity:
<font color = "red">                                                   ?
                   (sinx + cscx)² + (cosx + secx)² <font face = "symbol">º</font> tan²x + cot²x + 7
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We will work only with the left side, since
it looks more complicated than the right side.

Square the two binomials on the left
<font color = "red">                                                   ?
   sin²x+2sinx·cscx+csc²x + cos²x+2cosx·secx+sec²x <font face = "symbol">º</font> tan²x + cot²x + 7
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Replace the sin²x + cos²x by 1  
<font color = "red">                                                   ?
       1 + 2sinx·cscx + csc²x + 2cosx·secx + sec²x <font face = "symbol">º</font> tan²x + cot²x + 7
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Replace cscx by 1/sinx and secx by 1/cosx 
<font color = "red">                                                   ?
   1 + 2sinx·1/sinx + csc²x + 2cosx·1/cosx + sec²x <font face = "symbol">º</font> tan²x + cot²x + 7
<font color = "red">                                                   ?
                   1 + 2(1) + csc²x + 2(1) + sec²x <font face = "symbol">º</font> tan²x + cot²x + 7
<font color = "red">                                                   ?
                                 5 + csc²x + sec²x <font face = "symbol">º</font> tan²x + cot²x + 7
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Use identity csc²x <font face = "symbol">º</font> 1 + cot²x to replace csc²x
and sec² <font face = "symbol">º</font> 1 + tan²x to replace sec²x
<font color = "red">                                                   ?
                         5 + 1 + cot²x + 1 + tan²x <font face = "symbol">º</font> tan²x + cot²x + 7   
<font color = "red">                                                   ?
                                 7 + cot²x + tan²x <font face = "symbol">º</font> tan²x + cot²x + 7
<font color = "red">                                                   <font face = "symbol">Ö</font>     
<font color = "red">                                 tan²x + cot²x + 7 <font face = "symbol">º</font> tan²x + cot²x + 7
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Edwin</pre>