You can put this solution on YOUR website! Well to do this problem we need to remember a few facts:
1) sec(x) = 1/cos(x)
2) sin^2(x) = 1-cos^2(x)
3) tan(x) = sin(x)/cos(x)
Okay now to prove this identity we need to work with one side to see if we can get the other.
left hand side:
sinxtanx
= sinx(sinx/cosx)by number 3.
= sin^2x/cosx
= (1-cos^2x)/cosx by number 2.
= 1/cosx - cos^2x/cosx
= secx - cosx by number 1.
There now we found that the left hand side equals the right hand side and we're done. Just to show you that it doesn't matter what side we choose I'm going to work with the right hand side and get the left hand side as well.
Right hand side:
secx-cosx
=1/cosx - cosx by 1.
=(1-cos^2x)/cosx
=sin^2x/cosx by 2.
=sinx/cosx * sinx
=tanxsinx by 3.
Since , you can replace cos(x) in your equation to get:
This is the same as:
because .
Your expression has become:
Remove parentheses to get:
Since , you can replace in your equation to get:
Simplify to get:
Since this is what you wanted to prove, you're done.
Note I am showing sin^2(x) as sin(x)^2 because it doesn't come out good on the formula rendering routine with sin^2(x).
Example:
sin^2(x) shows up as
sin(x)^2 shows up as
The second version is technically incorrect but it shows up clearer so I used it.
Note that sin(x) * sin(x) really is (sin(x))^2 but I shortened it to sin(x)^2 to eliminate all those extra parentheses that muddied up the presentation.
Just remember that sin(x)^2 is the same as sin^2(x) and we'll be ok.
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