How do you simplify $cos (2 arcsin x)$ $cos(2 arcsin x)$ ?

Question

How do you use the properties of inverse trigonometric functions to evaluate $tan (arcsin (0.31))$ $tan(arcsin (0.31))$ ?
What is $sin ({sin}^{- 1} \frac{\sqrt{2}}{2})$ $\sin ( sin^{-1} frac{sqrt{2}}{2})$ ?
How do you find the exact value of $cos ({tan}^{- 1} \sqrt{3})$ $\cos(tan^{-1}sqrt{3})$ ?
How do you evaluate ${sec}^{- 1} \sqrt{2}$ $\sec^{-1} \sqrt{2}$ ?
How do you find $cos ({cot}^{- 1} \sqrt{3})$ $cos( cot^{-1} sqrt{3} )$ without a calculator?
How do you rewrite ${sec}^{2} ({tan}^{- 1} x)$ $sec^2 (tan^{-1} x)$ in terms of x?
How do you use the inverse trigonometric properties to rewrite expressions in terms of x?
How do you calculate ${sin}^{- 1} (0.1)$ $sin^-1(0.1)$ ?
How do you solve the inverse trig function ${cos}^{- 1} (- \frac{\sqrt{2}}{2})$ $cos^-1 (-sqrt2/2)$ ?
How do you solve the inverse trig function $sin ({sin}^{- 1} (\frac{1}{3}))$ $sin(sin^-1 (1/3))$ ?

36163 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-06-01T14:39:27

$cos (2 arcsin (x)) = 1 - 2 x^{2}$ $cos(2arcsin(x))=1-2x^2$

Using $cos (a + b) = cos (a) cos (b) - sin (a) sin (b)$ $cos(a+b)=Cos(a) Cos(b) - Sin(a) Sin(b)$ .

If $a = b$ $a = b$ then

$cos (2 a) = 1 - 2 {sin}^{2} (a)$ $cos(2a)=1-2sin^2(a)$

so

$cos (2 arcsin (x)) = 1 - 2 x^{2}$ $cos(2arcsin(x))=1-2x^2$

How do you simplify cos(2arcsinx)cos(2 arcsin x)?