Question

Sin square a by cos square a + cos square a by sin square a minus 1 by sin square a into cos square a is equal to minus 2

Answer 1

We have to prove that

             $\frac{sin^{2}A}{cos^{2}A}+\frac{cos^{2}A}{sin^{2}A}-\frac{1}{sin^{2}Acos^{2}A}=-2$

Proof:

Consider,

L.H.S $=\frac{sin^{2}A}{cos^{2}A}+\frac{cos^{2}A}{sin^{2}A}-\frac{1}{sin^{2}Acos^{2}A}$

       $=tan^{2}A+cot^{2}A-sec^{2}Acosec^{2}A$

                                [∵ 1/sinA = cosecA, 1/cosA = secA]

       $=tan^{2}A+cot^{2}A-(1+tan^{2}A)(1+cot^{2}A)$

                                 [∵$sec^{2}A=1+tan^{2}A,cosec^{2}A=1+cot^{2}A]$

       $=tan^{2}A+cot^{2}A-1-tan^{2}A-cot^{2}A-tan^{2}Acot^{2}A$

       $=-1-1$          [∵cotA = 1/tanA]

       $=-2$

       $=$  R.H.S

Answer 2

Answer:

$\begin{gathered}\frac{sin^{2}A}{cos^{2}A}+\frac{cos^{2}A}{sin^{2}A}-\frac{1}{sin^{2}Acos^{2}A}=-2\\\end{gathered}cos2Asin2A+sin2Acos2A−sin2Acos2A1=−2</p><p></p><p>Proof:</p><p></p><p>Consider,</p><p></p><p>L.H.S \begin{gathered}=\frac{sin^{2}A}{cos^{2}A}+\frac{cos^{2}A}{sin^{2}A}-\frac{1}{sin^{2}Acos^{2}A}\\\end{gathered}=cos2Asin2A+sin2Acos2A−sin2Acos2A1</p><p></p><p>       =tan^{2}A+cot^{2}A-sec^{2}Acosec^{2}A=tan2A+cot2A−sec2Acosec2A</p><p></p><p>                                [∵ 1/sinA = cosecA, 1/cosA = secA]</p><p></p><p>       =tan^{2}A+cot^{2}A-(1+tan^{2}A)(1+cot^{2}A)=tan2A+cot2A−(1+tan2A)(1+cot2A)</p><p></p><p>                                 [∵sec^{2}A=1+tan^{2}A,cosec^{2}A=1+cot^{2}A]sec2A=1+tan2A,cosec2A=1+cot2A]</p><p></p><p>       =tan^{2}A+cot^{2}A-1-tan^{2}A-cot^{2}A-tan^{2}Acot^{2}A=tan2A+cot2A−1−tan2A−cot2A−tan2Acot2A</p><p></p><p>       =-1-1=−1−1          [∵cotA = 1/tanA]</p><p></p><p>       =-2=−2</p><p></p><p>       ==  R.H.S</p><p></p><p>$

$\begin{gathered}\frac{sin^{2}A}{cos^{2}A}+\frac{cos^{2}A}{sin^{2}A}-\frac{1}{sin^{2}Acos^{2}A}=-2\\\end{gathered}cos2Asin2A+sin2Acos2A−sin2Acos2A1=−2</p><p></p><p>Proof:</p><p></p><p>Consider,</p><p></p><p>L.H.S \begin{gathered}=\frac{sin^{2}A}{cos^{2}A}+\frac{cos^{2}A}{sin^{2}A}-\frac{1}{sin^{2}Acos^{2}A}\\\end{gathered}=cos2Asin2A+sin2Acos2A−sin2Acos2A1</p><p></p><p>       =tan^{2}A+cot^{2}A-sec^{2}Acosec^{2}A=tan2A+cot2A−sec2Acosec2A</p><p></p><p>                                [∵ 1/sinA = cosecA, 1/cosA = secA]</p><p></p><p>       =tan^{2}A+cot^{2}A-(1+tan^{2}A)(1+cot^{2}A)=tan2A+cot2A−(1+tan2A)(1+cot2A)</p><p></p><p>                                 [∵sec^{2}A=1+tan^{2}A,cosec^{2}A=1+cot^{2}A]sec2A=1+tan2A,cosec2A=1+cot2A]</p><p></p><p>       =tan^{2}A+cot^{2}A-1-tan^{2}A-cot^{2}A-tan^{2}Acot^{2}A=tan2A+cot2A−1−tan2A−cot2A−tan2Acot2A</p><p></p><p>       =-1-1=−1−1          [∵cotA = 1/tanA]</p><p></p><p>       =-2=−2</p><p></p><p>       ==  R.H.S</p><p></p><p>$