Let c be the Champernowne's constant,
or c=0.123456789101112131415161718192021....

Show, without calculator aid, that

sin(c) + cos(c) + tan(c) > 10c

sin(c)>c-(c^3)/6

cos(c)>1-.5*c^2

tan(c)>sin(c)> c-(c^3)/6

The ">" sign results from ignoring the higher powers

in the Taylor's series

sin(c)+cos(c)+ tan(c)>1+2c-(.5*c^2+2(c^3)/6)=1+2c-d

d is a positive value ,slightly less than .008

1+2c=1.24691357...

let us see how this value is affected by subtracting .008 i.e. subtracting more than needed

1.24691357-0.008=1,2389135...while 10c=1.2345678

so sin(c)+cos(c)+ tan(c)>1+2c-d>10c

therefore

sin(c)+cos(c)+ tan(c)>1.2469135....>10c

SO

**sin(c)+cos(c)+ tan(c)> 10c**

q.e.d.

The above text was corrected and edited following a grave error

spotted by Charlie and broll.

Thanks, guys !

*Edited on ***December 13, 2014, 10:46 am**