Math, asked by pratyushadk1, 4 hours ago

Prove that: tan^4 A + sec^4 A = 1 + 2tan^2 A / cos^2 A

Answers

Answered by mathdude500
0

\large\underline{\sf{Solution-}}

Consider,

\sf \:  {tan}^{4}A +  {sec}^{4}A \\

can be rewritten as

\sf \:  =  \:  {sec}^{4}A +  {tan}^{4}A \\

\sf \:  =  \:  {( {sec}^{2}A) }^{2}+  {( {tan}^{2}A) }^{2} \\

We know,

\boxed{\sf \:  {x}^{2} +  {y}^{2} =  {(x - y)}^{2} + 2xy \: } \\

So, using this algebraic identity, we get

\sf \:  =  \:  {( {sec}^{2}A -  {tan}^{2}A ) }^{2} \: +  \: 2\:  {sec}^{2}A  \: {tan}^{2}A  \\

We know,

\boxed{\begin{aligned}& \qquad \:\sf \:  {sec}^{2}A -  {tan}^{2}A = 1  \qquad \: \\ \\& \qquad \:\sf \: secA= \dfrac{1}{cosA}  \end{aligned}} \qquad \\

So, using these Identities, we get

\sf \:  =  \:  {(1) }^{2}+ 2\times \dfrac{1}{ {cos}^{2} A} \times {tan}^{2}A  \\

\sf \:  =  \:  1+  \dfrac{2 {tan}^{2} A}{ {cos}^{2} A}   \\

Hence,

\implies\sf \: \boxed{\bf \: {tan}^{4}A +  {sec}^{4}A = 1+  \dfrac{2 {tan}^{2} A}{ {cos}^{2} A} \: } \\

\rule{190pt}{2pt}

Additional Information

\begin{gathered}\: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{{More \: Formulae}}}} \\ \\ \bigstar \: \bf{sinx =  \dfrac{1}{cosecx} }\\ \\ \bigstar \: \bf{cosx =  \dfrac{1}{secx} }\\ \\ \bigstar \: \bf{tanx = \dfrac{sinx}{cosx}  = \dfrac{1}{cotx} }\\ \\ \bigstar \: \bf{cot x= \dfrac{cosx}{sinx}  = \dfrac{1}{tanx} }\\ \\ \bigstar \: \bf{cosec x = \dfrac{1}{sinx} }\\ \\ \bigstar \: \bf{secx = \dfrac{1}{cosx} }\\ \\ \bigstar \: \bf{ {sin}^{2}x +  {cos}^{2}x = 1 } \\ \\ \bigstar \: \bf{ {sec}^{2}x -  {tan}^{2}x = 1  }\\ \\ \bigstar \: \bf{ {cosec}^{2}x -  {cot}^{2}x = 1 } \\ \\ \bigstar \: \bf{sin(90 \degree - x) = cosx}\\ \\ \bigstar \: \bf{cos(90 \degree - x) = sinx}\\ \\ \bigstar \: \bf{tan(90 \degree - x) = cotx}\\ \\ \bigstar \: \bf{cot(90 \degree - x) = tanx}\\ \\ \bigstar \: \bf{cosec(90 \degree - x) = secx}\\\: \end{array} }}\end{gathered}\end{gathered}\end{gathered} \\  \\

Answered by Aʙʜɪɪ69
0

Step-by-step explanation:

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