If A=0−tan⁡α/2tan⁡α/20 and I is a 2×2  unit matrix, then (I−A)cos⁡α−sin⁡αsin⁡αcos⁡α

# $\left(\mathrm{I}-\mathrm{A}\right)\left[\begin{array}{cc}\mathrm{cos}\mathrm{\alpha }& -\mathrm{sin}\mathrm{\alpha }\\ \mathrm{sin}\mathrm{\alpha }& \mathrm{cos}\mathrm{\alpha }\end{array}\right]$

1. A

$-\mathrm{I}+\mathrm{A}$

2. B

$\mathrm{I}+\mathrm{A}$

3. C

$-\mathrm{I}-\mathrm{A}$

4. D

None of these

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### Solution:

$\therefore \mathrm{I}-\mathrm{A}=\left[\begin{array}{cc}1& \mathrm{tan}\mathrm{\alpha }/2\\ -\mathrm{tan}\mathrm{\alpha }/2& 1\end{array}\right]$

$\begin{array}{l}=\left[\begin{array}{cc}1& \mathrm{tan}\mathrm{\alpha }/2\\ -\mathrm{tan}\mathrm{\alpha }/2& 1\end{array}\right]\left[\begin{array}{cc}\mathrm{cos}\mathrm{\alpha }& -\mathrm{sin}\mathrm{\alpha }\\ \mathrm{sin}\mathrm{\alpha }& \mathrm{cos}\mathrm{\alpha }\end{array}\right]\\ =\left[\begin{array}{cc}\mathrm{cos}\alpha +\mathrm{sin}\mathrm{\alpha tan}\mathrm{\alpha }/2& -\mathrm{sin}\mathrm{\alpha }+\mathrm{tan}\mathrm{\alpha }/2\mathrm{cos}\mathrm{\alpha }\\ -\mathrm{tan}\mathrm{\alpha }/2\mathrm{cos}\mathrm{\alpha }+\mathrm{sin\alpha }& \mathrm{cos}\alpha +\mathrm{sin}\mathrm{\alpha tan}\mathrm{\alpha }/2\end{array}\right]\\ =\left[\begin{array}{cc}\frac{\mathrm{cos\alpha cos\alpha }/2+\mathrm{sin}\mathrm{\alpha sin}\mathrm{\alpha }/2}{\mathrm{cos\alpha }/2}& \frac{\mathrm{sin}\mathrm{\alpha }/2\mathrm{cos}\mathrm{\alpha }-\mathrm{sin\alpha cos\alpha }/2}{\mathrm{cos\alpha }/2}\\ \frac{-\mathrm{sin}\mathrm{\alpha }/2\mathrm{cos}\mathrm{\alpha }+\mathrm{sin\alpha cos\alpha }/2}{\mathrm{cos\alpha }/2}& \frac{\mathrm{cos\alpha cos\alpha }/2+\mathrm{sin}\mathrm{\alpha sin}\mathrm{\alpha }/2}{\mathrm{cos\alpha }/2}\end{array}\right]\\ =\left[\begin{array}{cc}\frac{\mathrm{cos}\left(\alpha -\mathrm{\alpha }/2\right)}{\mathrm{cos\alpha }/2}& \frac{-\mathrm{sin}\left(\alpha -\mathrm{\alpha }/2\right)}{\mathrm{cos\alpha }/2}\\ \frac{\mathrm{sin}\left(\alpha -\mathrm{\alpha }/2\right)}{\mathrm{cos\alpha }/2}& \frac{\mathrm{cos}\left(\alpha -\mathrm{\alpha }/2\right)}{\mathrm{cos\alpha }/2}\end{array}\right]=\left[\begin{array}{cc}1& -\mathrm{tan}\mathrm{\alpha }/2\\ \mathrm{tan}\mathrm{\alpha }/2& 1\end{array}\right]\\ =\mathrm{I}+\mathrm{A}\end{array}$  +91

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