Question

Evaluate:
$\frac{s i n ^{2} 45°}{c o s e c ^{2} 30° - t a n ^{2} 45°}$

Approach

Evaluate $\frac{s i n ^{2} 4 5 ^{\circ}}{c o s e c ^{2} 3 0 ^{\circ} - t a n ^{2} 4 5 ^{\circ}}$ . Substitute known values: $sin 4 5^{\circ} = \frac{1}{2}$ , $cosec 3 0^{\circ} = 2$ , $tan 4 5^{\circ} = 1$ . Numerator: $(\frac{1}{2})^{2} = \frac{1}{2}$ . Denominator: $2^{2} - 1^{2} = 4 - 1 = 3$ . Thus, the expression equals $\frac{1/2}{3} = \frac{1}{6}$ .

Step-by-step working

1
Substitute values: $sin 4 5^{\circ} = \frac{1}{2}$ , $cosec 3 0^{\circ} = 2$ , $tan 4 5^{\circ} = 1$ .
2
Numerator: $sin^{2} 4 5^{\circ} = (\frac{1}{2})^{2} = \frac{1}{2}$ . Denominator: $cosec^{2} 3 0^{\circ} - tan^{2} 4 5^{\circ} = 2^{2} - 1^{2} = 4 - 1 = 3$ .
1.5 marks
3
Expression = $\frac{1/2}{3} = \frac{1}{6}$ .
0.5 marks

Concepts used

trigonometric ratioevaluatesimplifysin 45cosec 30tan 45

Approach

Step-by-step working

Concepts used

Similar question types