Question

Evaluate: (sin² 30° + 4cot² 45° - sec²60°)(cosec² 45° × sec²30°)​

Answer 1

To evaluate:-

• (sin² 30° + 4cot² 45° - sec²60°)(cosec² 45° × sec²30°)

Solution:-

(sin² 30° + 4cot² 45° - sec²60°)(cosec² 45° × sec²30°)

sin 30° = 1/2

cot 45° = 1

sec 60° = 2

cosec 45° = √2

sec 30° = 2/√3

Put all values,

$\implies \sf [ { (\dfrac{1}{2}) }^{2} + 4 \times {(1)}^{2} - {(2)}^{2}] \times [ {( \sqrt{2} )}^{2} \times {( \frac{2}{ \sqrt{3} } )}^{2}]$

$\implies \sf (\dfrac{1}{4} + \cancel{4} - \cancel{4}) \times (2 \times \frac{4}{3} )$

$\implies \sf \dfrac{1}{ \cancel{4} } \times 2 \times \dfrac{ \cancel{4}}{3}$

$\implies \sf \dfrac{2}{3}$

Therefore,

(sin² 30° + 4cot² 45° - sec²60°)(cosec² 45° × sec²30°) = $\sf \dfrac{2}{3}$

Answer 2

$\boxed{</p><p></p><p>\begin{minipage}{7 cm}</p><p></p><p>Fundamental Trigonometric Identities \\sin^2\theta + \cos^2\theta=1 \\ </p><p>1+\tan^2\theta = \sec^2\theta \\ </p><p>1+\cot^2\theta = \text{cosec}^2 \, \theta</p><p></p><p>\end{minipage}</p><p></p><p>}$