Question

plz answer this fastly I have to submit my assignment tommorow plz answer correctly​

Answer 1

Question:-

Evaluate:-

$\longrightarrow \sf \dfrac{ {sec}^{2}(90^{0} - \theta) - {cot}^{2} \theta }{2(sin^{2} {25}^{0} + {sin} {65}^{0}) } + \dfrac{2 {cos}^{2} {60}^{0} \times {tan}^{2} {28}^{0} \times {tan}^{2} {62}^{0} }{sin {30}^{0} }$

Solution:-

We have

$\sf \implies\dfrac{ {sec}^{2}(90^{0} - \theta) - {cot}^{2} \theta }{2(sin^{2} {25}^{0} + {sin} {65}^{0}) } + \dfrac{2 {cos}^{2} {60}^{0} \times {tan}^{2} {28}^{0} \times {tan}^{2} {62}^{0} }{sin {30}^{0} }$

$\sf \implies\dfrac{ {cosec}^{2} \theta - {cot}^{2} \theta }{2 \{sin^{2}( {90}^{0} - \ \: 65^{0} ) + {sin} {65}^{0} \} } + \dfrac{2 \times {cos}^{2} ( {90}^{0} - 30^{0} )\times {tan}^{2} ( {90}^{0} - {62}^{0}) \times {tan}^{2} {62}^{0} }{sin {30}^{0} }$

•Using below trignometric identity

$\underline{\boxed{ \red{ \longrightarrow\sf \: cosec^{2} \theta - cot ^{2} \theta = 1 }}}$

We get

$\sf \implies\dfrac{ 1 }{2 \{sin^{2}( {90}^{0} - \ \: 65^{0} ) + {sin} {65}^{0} \} } + \dfrac{2 \times {cos}^{2} ( {90}^{0} - 30^{0} )\times {tan}^{2} ( {90}^{0} - {62}^{0}) \times {tan}^{2} {62}^{0} }{sin {30}^{0} }$

Now again using below trignometric identity , where the value of theta is any angle in trigonometry .

$\bigstar \longrightarrow \tt {sin}^{2}(90 ^{0} - \theta) = cos ^{2} \theta$

$\bigstar \longrightarrow \tt {cos}^{2}(90 ^{0} - \theta) = sin ^{2} \theta$

$\bigstar\longrightarrow \tt {tan}^{2}(90 ^{0} - \theta) = cot ^{2} \theta$

We have

$\sf \implies\dfrac{ 1 }{ 2{{ \{cos^{2} 65^{0} }}+ {sin} {65}^{0} \} } + \dfrac{2 \times \cancel{ {sin}^{2} 30^{0} } \times {cot}^{2}{62}^{0}\times {tan}^{2} {62}^{0} }{ \cancel{sin {30}^{0}} }$

Again using below trignometric identity , where theta is any angle in trigonometry

$\bigstar \longrightarrow \tt {sin}^{2} \theta+ cos ^{2} \theta = 1$

$\bigstar\longrightarrow \tt {tan}^{2} \theta \times cot^{2} \theta = 1$

$\implies \sf \: \dfrac{1}{2} + \dfrac{2}{1}$

$\red{ \boxed{ \blue{\boxed{ \orange{\implies \bf \dfrac{5}{2}} \: \: \: \: \: \: \: \: \: }}}}$