Prove that : `(secA+tanA)(1-sinA)=cosA`

Answer

`a to r; b to q; c to s; d to p`

Explanation

a. We have `(4 sec20^(@))/( cosec20^(@)) = (sin 20^(@))/(cos 20^(@)) ( 4cos 20^(@) +1)`
`" " = ( 2sin 40^(@) + sin 20^(@))/( cos 20^(@))`
`" " = ( sin 40^(@) + ( sin 40^(@)+ sin 20^(@)))/( cos 20^(@))`
`" " = (sin 40^(@) + 2 sin 30^(@) cos 10^(@))/(cos 20^(@))`
`" " = ( sin 40^(@) + sin 80^(@))/( cos 20^(@))`
`" " = ( 2sin 60 ^(@) cos 20^(@))/( cos 20^(@))`
`" " = 2 xx (sqrt3)/(2) = sqrt3`
b. Let `f(x) = ( 2 cos ^(2) x + 8 sin ^(2) x)/( 2 sin 2x ) , f(x) gt0`
`" "= (cotx + 4 tan x)/( 2)`
`" " = ((sqrt(cot x) - 2sqrt(tanx))^(2)))/( 2) + 2 `
`therefore f(x) |_(min) = 2`
c. `(8sin 40^(@)* sin 50^(@) * tan 10^(@))/( cos 80^(@)) = (4(2sin 50^(@)* sin 40^(@))sin10^(@))/( sin 10^(@) * cos 10^(@))`
`" " 4( cos 10^(@) - cos 90^(@)) sin 10^(@))/( sin 10^(@) * cos 10^(@)) = 4`
d. L.H.S = `( 2sin 6A)/( sin 2A)= ( 2sin 3(2A))/( sin 2A)`
`" " = (2[3 sin 2A - 4sin ^(3)2 A])/(sin 2A)`
`" " = 2 (3-4 sin ^(2) 2A)`
`" " =2 (3-2 ( 1-cos 4A))`
`" "= 2+4 cos 4A`
`therefore a =2, b =4 `
`therefore a^(2)//b =1`