Question

Differentiate the function w.r.t.x. : cos⁻¹ (1-sin²x)

Answer 1

Answer:

$\frac{d cos^{-1}( cos^{2}x)}{dx}=\frac{sin 2x}{\sqrt{1-cos^{4}x}}$

Step-by-step explanation:

formula used:

$\frac{d cos^{-1}f(x) }{dx}=-\frac{1}{\sqrt{1-f(x)^{2} } }.\frac{df(x)}{dx}$

we know that

$1-sin^{2} x= cos ^{2}x$

so put this value into the given differentiation function

$\frac{d cos^{-1}( cos^{2}x)}{dx}= \frac{-1}{\sqrt{1-(cos^{2}x) ^{2}}} .\frac{d cos^{2}x }{dx} \\ \\ \\ =\frac{-1}{\sqrt{1-cos^{4}x}} .(2 cos x)\frac{d cos x }{dx} \\ \\ \\ =\frac{-1}{\sqrt{1-cos^{4}x}} .(2 cos x)(-sin x)\\\\ \\ =\frac{sin 2x}{\sqrt{1-cos^{4}x}}$