SOLUTION: <pre><font size = 4><b> Prove that this equation is an identity: ? (sinx + cscx)² + (cosx + secx)² <font face = "symbol">º</font> tan²x + cot²x

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Question 65913: 

Prove that this equation is an identity:
                                ?  
(sinx + cscx)² + (cosx + secx)² º tan²x + cot²x + 7
Found 2 solutions by uma, Edwin McCravy:
Answer by uma(370)   (Show Source): You can put this solution on YOUR website!
Consider the left hand side.
(sinx + cosecx)^2 + (cosx + secx)^2
= sin^2 x + cosec^2 x + 2sinx cosecx + cos^2 x + sec^2 x + 2cosx secx
= (sin^2 x + cos^2 x) + (cosec^2 x + sec^2 x) + 2 + 2 [grouping and sinx*cosecx = 1, secx*cosx=1]
= 1 + (1 + cot^2 x + 1 + tan^2 x) + 4 [using 1 + tan^2 x = sec^2 x and so on]
= cot^2 x + cosec^2 x + 7
= Right hand side.
Hence the proof.
Good Luck!!!
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
Answer by Edwin McCravy(279)

Prove that this equation is an identity:
                                                   ?
                   (sinx + cscx)² + (cosx + secx)² º tan²x + cot²x + 7

We will work only with the left side, since
it looks more complicated than the right side.

Square the two binomials on the left
                                                   ?
   sin²x+2sinx·cscx+csc²x + cos²x+2cosx·secx+sec²x º tan²x + cot²x + 7

Replace the sin²x + cos²x by 1  
                                                   ?
       1 + 2sinx·cscx + csc²x + 2cosx·secx + sec²x º tan²x + cot²x + 7

Replace cscx by 1/sinx and secx by 1/cosx 
                                                   ?
   1 + 2sinx·1/sinx + csc²x + 2cosx·1/cosx + sec²x º tan²x + cot²x + 7
                                                   ?
                   1 + 2(1) + csc²x + 2(1) + sec²x º tan²x + cot²x + 7
                                                   ?
                                 5 + csc²x + sec²x º tan²x + cot²x + 7

Use identity csc²x º 1 + cot²x to replace csc²x
and sec² º 1 + tan²x to replace sec²x
                                                   ?
                         5 + 1 + cot²x + 1 + tan²x º tan²x + cot²x + 7   
                                                   ?
                                 7 + cot²x + tan²x º tan²x + cot²x + 7
                                                   Ö     
                                 tan²x + cot²x + 7 º tan²x + cot²x + 7

Edwin
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