SOLUTION: Can you please help me solve this trigonometric identities question: 1)Prove the following identity. a) sinx/1-sin^2 x= tanx/cosx Use LS=RS

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Question 204710: Can you please help me solve this trigonometric identities question: 1)Prove the following identity.
a) sinx/1-sin^2 x= tanx/cosx
Use LS=RS
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
One thing to keep in mind when working with trig functions is tah they can all be "converted" in sin and cos. And sin can be "converted" to cos and vice versa!?

When working on trig identities and you don't see an easier path, then "convert" other functions into sin and cos using:
  • 1+-+sin%5E2%28x%29+=+cos%5E2%28x%29
  • 1+-+cos%5E2%28x%29+=+sin%5E2%28x%29
  • tan%28x%29+=+sin%28x%29%2Fcos%28x%29
  • cot%28x%29+=+cos%28x%29%2Fsin%28x%29
  • sec%28x%29+=+1%2Fcos%28x%29
  • csc%28x%29+=+1%2Fsin%28x%29

sin%28x%29%2F%281-sin%5E2%28x%29%29+=+tan%28x%29%2Fcos%28x%29
So we can start by noticing that the denominator on the left matches the first identity above so we can replace it:
sin%28x%29%2F%28cos%5E2%28x%29%29+=+tan%28x%29%2Fcos%28x%29
On the right side we can use the third identity above to replace tan(x):
sin%28x%29%2F%28cos%5E2%28x%29%29+=+%28sin%28x%29%2Fcos%28x%29%29%2Fcos%28x%29
And now, with some Algebra, we can change the right side to look like the left side. Using the fact that dividing by something is the same as multiplying by the reciprocal, we can change the division by cos(x) into the multiplication by the reciprocal: 1/cos(x):
sin%28x%29%2F%28cos%5E2%28x%29%29+=+%28sin%28x%29%2Fcos%28x%29%29%2A%281%2Fcos%28x%29%29
which simplifies to
sin%28x%29%2F%28cos%5E2%28x%29%29+=+sin%28x%29%2F%28cos%5E2%28x%29%29