How do you simplify sec(arctan(4x))?

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 $sin ({sin}^{- 1} (\frac{1}{3}))$ $sin(sin^-1 (1/3))$ ?

22412 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-09-24T00:37:01

Refer to explanation

Set $t = arctan 4 x \Rightarrow tan t = 4 x = \frac{4 x}{1}$ $t=arctan4x=>tant=4x=(4x)/1$

Now imagine a right triangle with perpendicular sides $1, 4 x$ $1,4x$
hence from Pythagorean theorem the hypotenuse is

$c = \sqrt{1^{2} + {(4 x)}^{2}} \Rightarrow c = \sqrt{1 + 16 x^{2}}$ $c=sqrt(1^2+(4x)^2)=>c=sqrt(1+16x^2)$

Finally we have that

$sec (arctan 4 x) = sec (t) = \frac{\sqrt{1 + 16 x^{2}}}{1} = \sqrt{1 + 16 x^{2}}$ $sec(arctan4x)=sec(t)=(sqrt(1+16x^2))/1=sqrt(1+16x^2)$