Question

Answer the question which is in attachment ​

Answer 1

EXPLANATION.

Differentiate.

⇒ (x + cos x)/(tan x).

As we know that,

Formula of :

Quotient rule.

$\sf \implies \dfrac{d}{dx} \bigg[\dfrac{f(x)}{g(x)} \bigg] = \dfrac{g(x) \dfrac{d}{dx}[f(x)] - f(x) \dfrac{d}{dx} [g(x)] }{[g(x)]^{2} }$

Using this formula in the equation, we get.

$\sf \implies \dfrac{tan(x) \times \dfrac{d}{dx}(x + cos x) - (x + cosx ) \times \dfrac{d}{dx} (tan x)}{[tan x]^{2} }$

$\sf \implies \dfrac{tan(x) (1 - sin x) - sec^{2} x (x + cos x)}{tan^{2} x}$

$\sf \implies \dfrac{tan(x) - tan(x)sin(x) - sec^{2}x - (x)sec^{2}x (cosx) }{tan^{2}x }$

                                                                                                                       

MORE INFORMATION.

(1) = d(sin x)/dx = cos x.

(2) = d(cos x)/dx = - sin x.

(3) = d(tan x)/dx = sec²x.

(4) = d(cot x)/dx = - cosec²x.

(5) = d(sec x)/dx = sec x tan x.

(6) = d(cosec x)/dx = - cosec x cot x.