Question

please solve the following ​

Answer 1

$\large\underline\blue{\bold{ANSWER - a}}$

Evaluate

$\tt \: sin12\degree \:sin25\degree \: sin30\degree \: sec65\degree \: sec78\degree \: sin90 \degree$

Identity used :-

$(1)\rm \: \boxed{ \pink{ \bf \: sin(90\degree - x)\: = \tt \:cosx }}$

$(2)\rm \: \boxed{ \pink{ \bf \: sec(90\degree - x) \: = \tt \:cosecx }}$

$(3)\rm \: \boxed{ \pink{ \bf \: cosecx \: = \tt \: \dfrac{1}{sinx} }}$

$(4)\rm \: \boxed{ \pink{ \bf \: secx \: = \tt \:\dfrac{1}{cosx} }}$

$\large\underline\purple{\bold{Solution :- }}$

$\tt \: sin12\degree \:sin25\degree \: sin30\degree \: sec65\degree \: sec78\degree \: sin90 \degree$

$\tt \: = sin12\degree \:sin25\degree \: \times \dfrac{1}{2} \times \: sec(90\degree - 25\degree) \: sec(90\degree - 12\degree) \: \times 1$

$\tt \: = \dfrac{1}{2} \times sin12\degree \: sin25\degree \: cosec25\degree \: cosec12\degree$

$\tt \: = \dfrac{1}{2} \times sin25\degree \: \times sin12\degree \: \times \dfrac{1}{sin25\degree \:} \times \dfrac{1}{sin12\degree \:}$

$\tt \: = \dfrac{1}{2}$

$\large\underline\blue{\bold{ANSWER - b}}$

$\rm :\implies\:x \: = \dfrac{2 - \sqrt{3} }{2 + \sqrt{3} }$

$\rm :\implies\:x = \dfrac{2 - \sqrt{3}}{2 + \sqrt{3}} \: \times \: \dfrac{2 - \sqrt{3}}{2 - \sqrt{3}}$

$\rm :\implies\:x \: = \: \dfrac{ {(2 - \sqrt{3})}^{2} }{ {(2)}^{2} - {( \sqrt{3} )}^{2} }$

$\rm :\implies\:x \: = \: \dfrac{4 + 3 - 4 \sqrt{3} }{4 - 3}$

$\rm :\implies\: \boxed{ \pink{ \bf \: x \: = \tt \: 7 - 4 \sqrt{3} }}$

Now,

Consider,

$\rm :\implies\:y \: = \dfrac{2 + \sqrt{3} }{2 - \sqrt{3} }$

$\rm :\implies\:y = \dfrac{2 + \sqrt{3}}{2 - \sqrt{3}} \: \times \: \dfrac{2 + \sqrt{3}}{2 + \sqrt{3}}$

$\rm :\implies\:y \: = \: \dfrac{ {(2 + \sqrt{3})}^{2} }{ {(2)}^{2} - {( \sqrt{3} )}^{2} }$

$\rm :\implies\:y \: = \: \dfrac{4 + 3 + 4 \sqrt{3} }{4 - 3}$

$\rm :\implies\: \boxed{ \pink{ \bf \: y \: = \tt \:7 + 4 \sqrt{3} }}$

Now,

Consider,

$\rm :\implies\: {x}^{2} - {y}^{2}$

$\rm :\implies\: {(7 - 4 \sqrt{3} )}^{2} - {(7 + 4 \sqrt{3}) }^{2}$

$\because \: ({ \pink{ \tt \: {(x + y)}^{2} - {(x - y)}^{2} = 4xy\: }})$

$\rm :\implies\: - 4 \times 7 \times 4 \sqrt{3}$

$\rm :\implies\: - 112 \sqrt{3}$