Question

Check for the derivative of sin x with respect to cos x is –cot x or not. Then write 1 if true and 0 if false.

Answer 1

Hint – In this question let y = sin x and z = cos x, then find $\dfrac{{dy}}{{dx}}$ and $\dfrac{{dz}}{{dx}}$, use these obtained derivatives to find the value of $\dfrac{{dy}}{{dz}}$. If it comes out equal to –cot x, then the answer is 1 else 0.

Complete step-by-step solution -
Let y = sin x...................... (1)
And z = cos x........................ (2)
Now differentiate equation (1) w.r.t. x we have,
$\Rightarrow \dfrac{{dy}}{{dx}} = \dfrac{d}{{dx}}\sin x = \cos x$.................. (3)
Now differentiate equation (2) w.r.t. x we have,
$\Rightarrow \dfrac{{dz}}{{dx}} = \dfrac{d}{{dx}}\cos x = - \sin x$........................ (4)
Now divide equation (3) from equation (4) we have,
$\Rightarrow \dfrac{{\dfrac{{dy}}{{dx}}}}{{\dfrac{{dz}}{{dx}}}} = \dfrac{{\cos x}}{{ - \sin x}}$
Now as we know that (cos/sin) = cot so we have,
$\Rightarrow \dfrac{{dy}}{{dz}} = \dfrac{{\cos x}}{{ - \sin x}} = - \cot x$
So this is the required differentiation of sin x w.r.t. cos x.
And the required answer is (-cot x).
So the given statement is true.
So according to the question we have to write 1.
So this is the required answer.

Note – This method is used to find the derivative of one entity with respect to another entity by taking the derivative of individual entities, and is most commonly used to solve any derivative problem of this kind. It is advised to remember the derivative of basic trigonometric ratios like sin x, cos x, tan x, cot x as it helps saving a lot of time.