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

Answer

A

Explanation

a.` 37^(@) = 45^(@) - 8^(@)`
`rArr tan 37^(@) = (1-tan 8^(@))/( 1+tan 8^(@))`
`rArr 1+tan 37^(@) = (2)/(1+tan 8^(@))`
Similarly, `1+ tan 23^(@) = (2)/( 1+ tan 22^(@))`
`rArr ((1+tan 8^(@))(1+tan 37^(@)))/( (1+tan22^(@)) (1+tan23^(@))) =1 `
b. `2 sum ((cot A + cot B)/(tanA +tanB)) = 2 sum (cosA cos B)/(sin A sin B)`
`" " = 2 (sum cos A cosB sinC)/(sinA sinB sinC)`
`" " = 2 (cot A cot B cot C sum tan C)`
`" " = 2cotA cot B cot C(tan A tanB tanC)`
c. `|cos (cos 3x + 3 cos x )| = |cos ( 4 cos ^(3) x)| ge 0`
` therefore ` minimum value is 0.
d. ` 16 sin ""(pi)/(5) sin""(2pi)/(5) sin ""(3pi)/(5)sin""(4pi)/(5)`
`= 16 sin ""(pi)/(5) sin ""(2pi)/(5) sin (pi - (2pi)/(5)) sin (pi-(pi)/(5))`
` = 16 sin ^(2)""(pi)/(5) * sin^(2)""(2pi)/(5)`
`= 4[cos""(pi)/(5) - cos""(3pi)/(5)]^(2)`
`= 4[ cos ""(pi)/(5) + sin ""(pi)/(10)]^(2)`
`= 4 [(sqrt5+1)/(4) + (sqrt5-1)/(4)]^(2)=5`