Question

Without using tables find the value of

`4/3 tan^2 30^circ + sin^2 60^circ - 3 cos^2 60^circ + 3/4 tan^2 60^circ - 2 tan^2 45^circ`

Answer 1

Given: Expression: `4/3 tan^2 30^circ + sin^2 60^circ - 3 cos^2 60^circ + 3/4 tan^2 60^circ - 2 tan^2 45^circ`

Step-wise calculation:

`tan 30^circ = 1/sqrt(3)` 

⇒ `tan^2 30^circ = 1/3`

So, `4/3 xx tan^2 30^circ`

= `4/3 xx 1/3`

= `4/9`

`sin 60^circ = sqrt(3)/2`

⇒ `sin^2 60^circ = 3/4`

`cos 60^circ = 1/2`

⇒ `cos^2 60^circ = 1/4`

So, –3 × cos² 60° 

= `-3 xx 1/4` 

= `-3/4`

`tan 60^circ = sqrt(3)`

⇒ tan² 60° = 3

So, `3/4 xx tan^2 60^circ`

= `3/4 xx 3` 

= `9/4`

tan 45° = 1

⇒ tan² 45° = 1

So, –2 × tan² 45° = –2.

Now add the terms: `4/9 + 3/4 - 3/4 + 9/4 - 2`

The `+3/4` and `-3/4` cancel, leaving:

`4/9 + 9/4 - 2`

= `4/9 + (9/4 − 8/4)` 

= `4/9 + 1/4`

Common denominator 36:

`(16 + 9)/36 = 25/36`