Question

Sanjay had spend 590000+5646516​

Answer 1

Step-by-step explanation:

ToProve★(1+cot2A)(1+cosA)(1−cosA)=1

Taking L.H.S

\rm \longrightarrow \: (1 + \cot {}^{2} A)(1 - \cos \:A )(1 + \cos \: A)⟶(1+cot2A)(1−cosA)(1+cosA)

\begin{gathered}\underline{ \boxed{ \sf{Using \: identity \: = (x + y)(x - y) = ( {x}^{2} - {y}^{2} )}}} \\ \\ \rm \longrightarrow \: ( 1 + \cot {}^{2} A)(1 - \cos {}^{2} A)\end{gathered}Usingidentity=(x+y)(x−y)=(x2−y2)⟶(1+cot2A)(1−cos2A)

\begin{gathered}\underline{\frak{We \: know \: that \: }}\begin{cases} \sf \bullet \: 1 + cot {}^{2} (A) = csc {}^{2} (A) \\ \sf \bullet \: 1 - cos {}^{2} (A) = sin {}^{2} (A) \\ \sf \: \bullet \: csc (A) = \dfrac{1}{sin(A)} \end{cases}\end{gathered}Weknowthat⎩⎪⎪⎪⎨⎪⎪⎪⎧∙1+cot2(A)=csc2(A)∙1−cos2(A)=sin2(A)∙csc(A)=sin(A)1

\begin{gathered}\rm{ \longrightarrow \: csc {}^{2} A \times \sin \: {}^{2} A } \\ \\ \rm \longrightarrow \: \dfrac{1}{ ( \cancel{\sin {}^{2} \:A ) }} \times \cancel{\sin \: {}^{2} A} \\ \\ \rm \longrightarrow \: 1\end{gathered}⟶csc2A×sin2A⟶(sin2A)1×sin2A⟶1

L.H.S = R.H.S ( HENCE PROVED )

ToProve★(1+cot2A)(1+cosA)(1−cosA)=1

Taking L.H.S

\rm \longrightarrow \: (1 + \cot {}^{2} A)(1 - \cos \:A )(1 + \cos \: A)⟶(1+cot2A)(1−cosA)(1+cosA)

\begin{gathered}\underline{ \boxed{ \sf{Using \: identity \: = (x + y)(x - y) = ( {x}^{2} - {y}^{2} )}}} \\ \\ \rm \longrightarrow \: ( 1 + \cot {}^{2} A)(1 - \cos {}^{2} A)\end{gathered}Usingidentity=(x+y)(x−y)=(x2−y2)⟶(1+cot2A)(1−cos2A)

\begin{gathered}\underline{\frak{We \: know \: that \: }}\begin{cases} \sf \bullet \: 1 + cot {}^{2} (A) = csc {}^{2} (A) \\ \sf \bullet \: 1 - cos {}^{2} (A) = sin {}^{2} (A) \\ \sf \: \bullet \: csc (A) = \dfrac{1}{sin(A)} \end{cases}\end{gathered}Weknowthat⎩⎪⎪⎪⎨⎪⎪⎪⎧∙1+cot2(A)=csc2(A)∙1−cos2(A)=sin2(A)∙csc(A)=sin(A)1

\begin{gathered}\rm{ \longrightarrow \: csc {}^{2} A \times \sin \: {}^{2} A } \\ \\ \rm \longrightarrow \: \dfrac{1}{ ( \cancel{\sin {}^{2} \:A ) }} \times \cancel{\sin \: {}^{2} A} \\ \\ \rm \longrightarrow \: 1\end{gathered}⟶csc2A×sin2A⟶(sin2A)1×sin2A⟶1

L.H.S = R.H.S ( HENCE PROVED )

Answer 2

5646516+590000

= 6236516

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