SOLUTION: Verify the identity. cotangent of x to the second power divided by quantity cosecant of x plus one equals quantity one minus sine of x divided by sine of x

Algebra ->  Trigonometry-basics -> SOLUTION: Verify the identity. cotangent of x to the second power divided by quantity cosecant of x plus one equals quantity one minus sine of x divided by sine of x      Log On


   

Question 942179: Verify the identity.
cotangent of x to the second power divided by quantity cosecant of x plus one equals quantity one minus sine of x divided by sine of x
Answer by MathLover1(20850) About Me  (Show Source):
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Verify the identity.
cotangent of x to the second power :%28ctg%28x%29%29%5E2
divided by quantity cosecant of x plus one :%28csc%28x%29%2B1%29
so far we have %28ctg%5E2%28x%29%29%2F%28csc%28x%29%2B1%29
then, that equals %28ctg%5E2%28x%29%29%2F%28csc%28x%29%2B1%29=
to quantity one minus sine of x: %281-sin%28x%29%29
divided by sine of x: %281-sin%28x%29%29%2Fsin%28x%29
so, we have:
%28ctg%5E2%28x%29%29%2F%28csc%28x%29%2B1%29=%281-sin%28x%29%29%2Fsin%28x%29 ..............since 1-sin%28x%29+=+cot%5E2%28x%29 and cot%5E2%28x%29=%28cos%28x%29%29%5E2%2F%28sin%28x%29%29%5E2, we have



or





.............since cos%28x%29%2Fsin%28x%29=ctg%28x%29


ctg%5E2%28x%29%2F%28csc%28x%29%2B1%29=%28cos%28x%29+cot%28x%29%29%2F%28sin%28x%29%2B1%29 ................since1%2Bsin%28x%29+=+1%2B1%2F%28csc%28x%29%29





............csc%28x%29cos%28x%29=ctg%28x%29



ctg%5E2+%28x%29%2F%28csc%28x%29%2B1%29=cot%5E2%28x%29%2F%28%28csc%28x%29%2B1%29%29%29