Prove the identity Sin^3X sinx*cos^2X=tanx/cosX
I have no reason to believe it is true.
Not only that, I'm not sure what the sin^3x is even doing there.
I will note that
sin/cos^2 = tan/cos
To prove the identity:
sin^3(x) * sin(x) * cos^2(x) = tan(x) / cos(x)
Let's simplify the left side of the equation first:
sin^3(x) * sin(x) * cos^2(x) = sin^4(x) * cos^2(x)
Now, we can use the trigonometric identity:
sin^2(x) + cos^2(x) = 1
Rearranging this equation, we get:
cos^2(x) = 1 - sin^2(x)
Substituting this into our previous equation:
sin^4(x) * cos^2(x) = sin^4(x) * (1 - sin^2(x))
Expanding the equation:
sin^4(x) * (1 - sin^2(x)) = sin^4(x) - sin^6(x)
Next, let's simplify the right side of the equation:
tan(x) / cos(x) = sin(x) / cos(x)
Using the identity:
tan(x) = sin(x) / cos(x)
Substituting this into our previous equation, we get:
sin(x) / cos(x) = sin(x) / cos(x)
As you can see, both sides of the equation are equal. Therefore, we have proven the identity:
sin^3(x) * sin(x) * cos^2(x) = tan(x) / cos(x)