Question

please help me!!!!!​

Answer 1

GIVEN :-

$\rm{ \dfrac{sin \: a + cos \: a}{sin \: a - cos \: a} \: + \: \dfrac{sin \: a - cos \: a}{sin \: a + cos \: a} }$

TO FIND :-

$\rm{ value \: of \: \: \dfrac{sin \: a + cos \: a}{sin \: a - cos \: a} \: + \: \dfrac{sin \: a - cos \: a}{sin \: a + cos \: a} }$

SOLUTION :-

FIRST WE WILL ADD THE TERMS

$\implies \boxed{\rm{ \dfrac{a}{b} + \dfrac{b}{a} } = \dfrac{ {a}^{2} + {b}^{2} }{ab} }$

$\implies \rm{ \bf{where \: a = sin \: a + cos \: a \:} }$

$\implies \rm{ \bf where \: b \: = sin \: a - cos \: a \:}$

NOW PUT THE VALUE OF A and B

$\implies \rm{ \dfrac{(sin \: a + cos \: a) ^{2} \: +( {sin \: a - cos \: a})^{2} }{(sin \: a + cos \: a)(sin \: a - cos \: a)} }$

$\implies \boxed{\rm{ ( a + b)(a - b) = {a}^{2} - {b}^{2} }}$

$\implies \rm{ \bf{where \: a = sin \: a + cos \: a \:} }$

$\implies \rm { \bf{where \: b \: = sin \: a - cos \: a \:}}$

$\implies \rm{ \dfrac{(sin \: a + cos \: a) ^{2} \: +( {sin \: a - cos \: a})^{2} }{(sin ^{2} a \ - cos ^{2} \: a)} }$

$\implies \boxed{ \rm{ {(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab }}$

$\implies \boxed{ \rm{ {(a - b)}^{2} = {a}^{2} + {b}^{2} - 2ab }}$

NOW , ( a + b )² + ( a - b )² :-

$\implies \rm{ ( {a}^{2} + {b}^{2} + 2ab ) + ({a}^{2} + {b}^{2} - 2ab })$

$\implies \rm{ 2{a}^{2} + 2 {b}^{2} }$

$\implies \boxed{ \rm2( {a}^{2} + {b}^{2} ) }$

$\implies \rm{ \bf{where \: a = sin \: a} }$

$\implies \rm{ \bf{where \: b= cos \: a} }$

PUT THE VALUE OF A and B

$\implies \rm{ \dfrac{2(sin \: ^{2} a + cos \: ^{2} \: a)}{(sin ^{2} a \ - cos ^{2} \: a)} }$

$\implies \boxed{ \rm{ {sin}^{2} a + {cos}^{2}a } = 1}$

$\implies \rm{ \dfrac{2(1)}{(sin ^{2} a \ - cos ^{2} \: a)} }$

$\implies \boxed{ \boxed{ \rm{ \dfrac{2}{(sin ^{2} a \ - cos ^{2} \: a)} }}}$

OTHER INFORMATION :-

TRIGNOMETRIC IDENTITIES :-

• SIN²∅ + COS²∅ = 1
• SEC²∅ - TAN²∅ = 1
• COSEC²∅ - COT²∅ = 1

TRIGNOMETRIC COMPLEMENTARY ANGLES :-

• SIN 90 = COS ( 90 - ∅ )
• COS 90 = SIN ( 90 - ∅ )
• TAN 90 = COT ( 90 - ∅ )
• COT 90 = TAN ( 90 - ∅ )
• COSEC 90 = SEC ( 90 - ∅ )
• SEC 90 = COSEC ( 90 - ∅ )

TRIGNOMATERIC RATIOS :-

• SIN ∅ = 1/ COSEC ∅
• COS ∅ = 1/ SEC ∅
• TAN ∅ = 1/ COT ∅
• TAN ∅ = SIN ∅ / COS ∅
• COT ∅ = COS ∅ / SIN ∅