SOLUTION: <pre><font size = 4><b> Prove that this equation is an identity: ? (sinx + cscx)� + (cosx + secx)� <font face = "symbol">�</font> tan�x + cot�x

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Question 65913:
Prove that this equation is an identity: ? (sinx + cscx)� + (cosx + secx)� � tan�x + cot�x + 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)� + (cosx + secx)� � tan�x + cot�x + 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 ? sin�x+2sinx�cscx+csc�x + cos�x+2cosx�secx+sec�x � tan�x + cot�x + 7 Replace the sin�x + cos�x by 1 ? 1 + 2sinx�cscx + csc�x + 2cosx�secx + sec�x � tan�x + cot�x + 7 Replace cscx by 1/sinx and secx by 1/cosx ? 1 + 2sinx�1/sinx + csc�x + 2cosx�1/cosx + sec�x � tan�x + cot�x + 7 ? 1 + 2(1) + csc�x + 2(1) + sec�x � tan�x + cot�x + 7 ? 5 + csc�x + sec�x � tan�x + cot�x + 7 Use identity csc�x � 1 + cot�x to replace csc�x and sec� � 1 + tan�x to replace sec�x ? 5 + 1 + cot�x + 1 + tan�x � tan�x + cot�x + 7 ? 7 + cot�x + tan�x � tan�x + cot�x + 7 � tan�x + cot�x + 7 � tan�x + cot�x + 7 Edwin