Ex 3.2, 13 - Chapter 3 Class 12 Matrices (Term 1)

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Ex 3.2, 13 If F (x) = [■8(cos⁡𝑥&〖−sin〗⁡𝑥&[email protected]⁡𝑥&cos⁡𝑥&[email protected]&0&1)] , Show that F(x) F(y) = F(x + y) We need to show F(x) F(y) = F(x + y) Taking L.H.S. Given F(x) = [■8(cos⁡𝑥&〖−sin〗⁡𝑥&[email protected]⁡𝑥&cos⁡𝑥&[email protected]&0&1)] Finding F(y) Replacing x by y in F(x) F(y) = [■8(cos⁡𝑦&〖−sin〗⁡𝑦&[email protected]⁡𝑦&cos⁡𝑦&[email protected]&0&1)] Now, F(x) F(y) = [■8(cos⁡𝑥&〖−sin〗⁡𝑥&[email protected]⁡𝑥&cos⁡𝑥&[email protected]&0&1)] [■8(cos⁡𝑦&〖−sin〗⁡𝑦&[email protected]⁡𝑦&cos⁡𝑦&[email protected]&0&1)] = [■8(cos⁡𝑥 cos⁡𝑦+(〖−sin〗⁡𝑥 ) sin⁡〖𝑦+0 〗 &cos⁡〖𝑥(−sin⁡〖𝑦)+(−sin⁡〖𝑥)〖cos y〗⁡〖+ 0〗 〗 〗 〗&0+0+0×[email protected]⁡〖𝑥 cos⁡〖𝑦+cos⁡〖𝑥 sin⁡〖𝑦+0〗 〗 〗 〗&sin⁡𝑥 (−sin⁡〖𝑦)+〗 cos⁡〖𝑥 cos⁡〖𝑦+0〗 〗&0+0+0×[email protected]×cos⁡〖𝑦 +0×sin⁡〖𝑦+0×1〗 〗&0×(−sin⁡〖𝑦)+0×cos⁡〖𝑦+0〗 〗&0+0+1×1)] = [■8(cos⁡𝑥 cos⁡𝑦 〖−sin〗⁡𝑥.sin⁡〖𝑦 〗 &〖−cos〗⁡〖𝑥 sin⁡〖𝑦−sin⁡〖𝑥 cos⁡𝑦 〗 〗 〗&[email protected]⁡〖𝑥 cos⁡〖𝑦+cos⁡〖𝑥 sin⁡𝑦 〗 〗 〗&−sin⁡𝑥 sin⁡〖𝑦+〗 cos⁡〖𝑥 cos⁡𝑦 〗&[email protected]&0&1)] We know that cos x cos y – sin x sin y = cos (x + y) & sin x cos y + cos x sin y = sin (x + y) = [■8(cos⁡〖(𝑥+𝑦)〗 &〖−[cos〗⁡〖𝑥 sin⁡〖𝑦+sin⁡〖𝑥 cos⁡〖𝑦]〗 〗 〗 〗&[email protected]⁡〖(𝑥+𝑦)〗&cos⁡𝑥 cos⁡〖𝑦 −〗 sin⁡〖𝑥 sin⁡𝑦 〗&[email protected]&0&1)] = [■8(cos⁡〖(𝑥+𝑦)〗 &−sin⁡〖(𝑥+𝑦)〗&[email protected]⁡〖(𝑥+𝑦)〗&cos⁡〖(𝑥+𝑦)〗&[email protected]&0&1)] Taking R.H.S F(x + y) Replacing x by (x + y) in F(x) = [■8(cos⁡〖(𝑥+𝑦)〗 &−sin⁡〖(𝑥+𝑦)〗&[email protected]⁡〖(𝑥+𝑦)〗&cos⁡〖(𝑥+𝑦)〗&[email protected]&0&1)] = L.H.S. Hence proved