Question

Prove that `cos^2 30^circ + sin 30^circ + tan^2 45^circ = 2 1/4`.

Answer 1

1. Identify trigonometric values

First, we recall the exact values for the trigonometric functions at the given angles:

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

`sin 30^circ = 1/2`

tan 45° = 1

2. Substitute and Square

Next, substitute these values into the left-hand side (LHS) of the equation:

LHS = `(sqrt(3)/2)^2 + (1/2) + (1)^2`

Calculate the squares:

`(sqrt(3)/2)^2 = 3/4`

(1)² = 1

Now the expression looks like this:

LHS = `3/4 + 1/2 + 1`

3. Simplify the expression

To add these terms, convert them to a common denominator of 4:

`1/2 = 2/4`

`1 = 4/4`

Combine the fractions:

LHS = `3/4 + 2/4 + 4/4`

LHS = `(3 + 2 + 4)/4 = 9/4`

4. Convert to mixed number

Finally, convert the improper fraction `9/4` into a mixed number:

`9/4 = 2` with a remainder of 1

⇒ `2 1/4`

Since the LHS equals the RHS, the statement is proved.