Question

Evaluate the following
(i) `sin 60^@ cos30^@ + sin 30^@ cos 60^@`
(ii) `2tan^2 45^@ + cos^2 30^@ - sin^2 60^@`
(iii) `(cos45^@)/(sec30^@ + c o s e c 30^@)`
(iv) `(sin30^@ + tan45^@ - c o s e c 60^@ )/(sec30^@ + cos60^@ +cot45^@)`
(v) `(5cos^2 60^@ + 4sec^2 30^@ - tan^2 45^@ )/ (sin^2 30^@ + cos^2 30^@ ) `

Answer 1

(i) `sin60^@cos30^@ + sin30^@cos60^@`
`=sin(60^@+30^@) = sin90^@ = 1`

(ii)`2tan^2 45^@ +cos^2 30^@ - sin^2 60^@`
`=2(1)^2 +(sqrt3/2)^2 - (sqrt3/2)^2`
`=2`

(iii)`cos45^@/(sec30^@+cosec30^@)`
`= (1/sqrt2)/(2/sqrt3+2)`
`=(sqrt3)/(2sqrt2(1+sqrt3))`

(iv)`(sin30^@+tan45^@-cosec60^@)/(sec30^@+cos60^@+cot45^@)`
`=(1/2+1 - 2/sqrt3)/(2/sqrt3+1/2+1)`
`=(sqrt3+2sqrt3-4)/(4+sqrt3+2sqrt3)`
`=(3sqrt3-4)/(3sqrt3+4)`

(v)`(5cos^2 60^@+4sec^2 30^@ - tan^2 45^@)/(sin^2 30^@+cos^2 30^@)`
`=(5(1/2)^2 +4(2/sqrt3)^2 - 1^2)/((1/2)^2 +(sqrt3/2)^2)`
`=(5/4+16/3-1)/(1/4+3/4)`
`=((15+64-12)/12)/1`
`= 67/12.`