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Prove that :
            \left ( \sin \theta +1+\cos \theta \right )\left ( \sin \theta -1+\cos \theta \right )\cdot \sec \theta \, cosec\, \theta = 2

 

 

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Prove \rightarrow \left ( \sin \theta +1+\cos \theta \right )\left ( \sin \theta -1+\cos \theta \right )\cdot \sec \theta \, cosec\, \theta = 2
Proof \rightarrow Taking L.H.S
\Rightarrow \left ( \left \{ \sin \theta +\cos \theta \right \}+1 \right )\left ( \left \{ \sin \theta +\cos \theta \right \} -1\right )-\sec \theta \, cosec\, \theta  \Rightarrow \left ( \left ( \sin \theta +\cos \theta \right )^{2}-1^{2} \right )\sec \theta \, cosec\, \theta   \left ( \therefore \left ( a+b \right )\left ( a-b \right )= a^{2}-b^{2} \right )
\Rightarrow \left ( \sin ^{2}\theta +\cos ^{2}\theta +2\sin \theta \cos \theta -1 \right )\sec \theta \, cosec\, \theta \; \; \left ( \therefore \left ( a+b \right )^{2} = a^{2}+b^{2}+2ab\right )\Rightarrow \left ( 1+2\sin \theta \cos \theta -1 \right )\sec \theta \cdot cosec\, \theta \: \: \left ( \therefore \sin ^{2}\theta +\cos ^{2}\theta = 1 \right )\Rightarrow 2\sin \theta \cos \theta \sec \theta\, cosec\, \theta
\Rightarrow 2\sin \theta \cos \theta \times \frac{1}{\cos \theta }\times \frac{1}{\sin \theta }               \begin{bmatrix} \therefore \frac{1}{\cos \theta }= \sec \theta \\ \frac{1}{\sin \theta }= cosec\, \theta \end{bmatrix}
\Rightarrow 2                                                                                
L.H.S = R.H.S
Hence Proved             

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Safeer PP

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