Question

Evaluate: `(5 sin^2 45^circ - 3 tan^2 30^circ)/(2 sec^2 30^circ)`

Answer 1

1. Identify trigonometric values

First, we recall the standard values for the trigonometric functions involved:

`sin 45^circ = 1/sqrt(2)`

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

`sec 30^circ = 1/(cos 30^circ)`

= `1/(sqrt(3)/2)`

= `2/sqrt(3)`

2. Substitute and square values

Now, substitute these values into the expression and calculate the squares:

`sin^2 45^circ = (1/sqrt(2))^2`

= `1/2`

`tan^2 30^circ = (1/sqrt(3))^2`

= `1/3`

`sec^2 30^circ = (2/sqrt(3))^2`

= `4/3`

3. Simplify the numerator

Substitute the squared values into the numerator:

Numerator = `5(1/2) - 3(1/3)`

Numerator = `5/2 - 1`

= `(5 - 2)/2`

= `3/2`

4. Simplify the denominator

Substitute the squared value into the denominator:

Denomiantor = `2(4/3)`

= `8/3`

5. Calculate the final result

Divide the simplified numerator by the simplified denominator:

Expression = `(3/2)/(8/3)`

To divide fractions, multiply by the reciprocal:

Expression = `3/2 xx 3/8`

= `9/16`