Question

Evaluate the following:

`sin^2 30^circ cos^2 45^circ + 4tan^2 30^circ + 1/2 sin^2 90^circ + 1/8 cot^2 60^circ`

Answer 1

Given: `sin^2 30^circ cos^2 45^circ + 4tan^2 30^circ + 1/2 sin^2 90^circ + 1/8 cot^2 60^circ`.

`sin 30^circ = 1/2`

⇒ `sin^2 30^circ = 1/4`

`cos 45^circ = sqrt(2)/2`

⇒ `cos^2 45^circ = 1/2`

⇒ `sin^2 30^circ cos^2 45^circ = (1/4)(1/2) = 1/8`

`tan 30^circ = 1/sqrt(3)`

⇒ `tan^2 30^circ = 1/3`

⇒ `4 tan^2 30^circ = 4 × (1/3) = 4/3`

sin 90° = 1

⇒ `(1/2) sin^2 90^circ = 1/2`

`cot 60^circ = 1/sqrt(3)`

⇒ `cot^2 60^circ = 1/3`

⇒ `(1/8) cot^2 60^circ = (1/8) × (1/3) = 1/24`

Now add: `1/8 + 4/3 + 1/2 + 1/24`

Convert to common denominator 24:

`3/24 + 32/24 + 12/24 + 1/24`

= `(3 + 32 + 12 + 1)/24` 

= `48/24`

= 2