Question

Question #c4d83

Answer 1

It may depend on the angular units. IF my conversions are correct.

Explanation:

First, convert the inverse functions into the “regular” function.
`cos(tan^(-)1 (1/4) + cos^-1 (1/2))`

`tan^(-1) (¼) = tan 4` and `cos^(-1) (½) = cos 2`

`cos(tan(4) + cos(2))` ; `tan phi = ((sin phi)/(cos phi))`

`cos((sin(4)/(cos(4)) + cos(2))` ; `(sin(4)) + cos^2(2))`

From here it depends on whether those are degrees or radians. Without an even multiplier of pi, I'm not sure what "without a calculator" means. SOMETHING has to calculate it.

If any of the previous conversions are incorrect, my apologies, and a reviewer may delete this answer (or let me know, and I will).

Answer 2

see below

Explanation:

From the `color(red)(tan^-1(1/4)` we have the side `color(red)(opposite)` to `color(red)(angle theta=1` and the `color(red)( adjacent` to `color(red)(angle theta= 4`. Now lets find the `color(red)(hypote n use` using pythagorean theorem. That is,

`color(red)(c=sqrt(1^2 +4^2 )=sqrt17`. Hence, `color(blue)(cos(tan^-1(1/4))=cos theta=(adjacent)/(hypote n use)=4/sqrt17=(4sqrt17)/17`

`color(blue)( sin( tan^-1(1/4))=sin theta=(opposite)/(hypote n use)=1/sqrt17=(sqrt17)/17`

From the `color(orange)(cos^-1 (1/2)` we have the side `color(orange)(adjacent)` to`color(orange)( angle theta = 1` and the `color(orange)(hypote n use = 2`. So this is a `color(orange)(30^@-60^@-90^@` triangle and therefore the `color(orange)(opposite)` to `color(orange)(angle theta=sqrt3`. You can check by using the pythagorean theorem. Thus,

`color(blue)(cos(cos^-1(1/2))=cos theta = (adjacent)/(hypote n use) =1/2`

`color(blue)(sin(cos^-1(1/2))=sin theta = (opposite)/(hypote n use) = sqrt3/2`

Then using the property `color(blue) (cos(A+B)=cosAcosB-sinAsinB)`

we have

`color(blue)(A=tan^- 1 (1/4) and B=cos^(-1)(1/2)`
Then

`cos(tan^-1 (1/4)+cos^-1 (1/2))=color(magenta)(cos(tan^-1(1/4))cos(cos^-1(1/2))-sin(tan^-1(1/4))sin(cos^-1(1/2))`

`color(magenta)(=(4sqrt17)/17 *(1/2)-sqrt17/17*sqrt3/2)`

`color(magenta)(=(4sqrt17)/34-sqrt51/34`

`color(blue)( :.=(4sqrt17-sqrt51)/34`