Prove the identity Sin^3X+sinx*cos^2X=tanx/cosX
To prove the identity sin^3(x) + sin(x) * cos^2(x) = tan(x) / cos(x), we'll work with only one side of the equation and manipulate it until we end up with the other side of the equation.
Starting with the left side of the equation, which is sin^3(x) + sin(x) * cos^2(x), we can rewrite sin^3(x) as (sin(x))^3. Then, using the identity sin^2(x) = 1 - cos^2(x), we can substitute it into the equation:
(sin(x))^3 + sin(x) * cos^2(x)
= (sin(x))^3 + sin(x) * (1 - sin^2(x))
= (sin(x))^3 + sin(x) - sin(x) * sin^2(x)
Next, we can factor out sin(x) from the final two terms:
(sin(x))^3 + sin(x) - sin(x) * sin^2(x)
= sin(x) * [(sin(x))^2 + 1 - sin^2(x)]
Now, we can simplify the expression within the square brackets:
(sin(x))^2 + 1 - sin^2(x)
= sin^2(x) + 1 - sin^2(x)
= 1
Therefore, sin^3(x) + sin(x) * cos^2(x) simplifies to sin(x).
Comparing this simplified expression to the right side of the equation, which is tan(x) / cos(x), we can see that they are equal. Thus, we have proven the identity sin^3(x) + sin(x) * cos^2(x) = tan(x) / cos(x).
factor left
sinX)(sin^2x+cos^2X)=
sinX*1 which does not equal the right.
Lets look at the original idenity:
at x=45 deg
.707^3+.707(.707^2)=
.707^3 * 2= .706
now, the right side for 45deg
1/.707=1.4
The indenity is not an idenity as you have written it.