| If \[p=a\sin x+b\cos x\]and \[q=a\cos x-b\sin x,\] then what is the value of \[{{p}^{2}}+{{q}^{2}}?\] |
A) \[a+b\]
B) \[ab\]
C) \[{{a}^{2}}+{{b}^{2}}\]
D) \[{{a}^{2}}-{{b}^{2}}\]
Correct Answer: C
Solution :
| Given, \[p=a\sin x+b\cos x\] ...(i) |
| and \[q=a\cos x-b\sin x\] ...(ii) |
| On squaring Eqs. (i) and (ii), we get |
| \[{{p}^{2}}={{a}^{2}}{{\sin }^{2}}x+{{b}^{2}}{{\cos }^{2}}x+2ab\sin x\cos x\] |
| and \[{{q}^{2}}={{a}^{2}}{{\cos }^{2}}x+{{b}^{2}}{{\sin }^{2}}x-2ab\sin x\cos x\] |
| Now,\[{{p}^{2}}+{{q}^{2}}={{a}^{2}}{{\sin }^{2}}x+{{b}^{2}}+{{\cos }^{2}}x+2ab\sin x\cos x\] |
| \[+\,\,{{a}^{2}}{{\cos }^{2}}+{{b}^{2}}{{\sin }^{2}}x-2ab\sin x\cos x\] |
| \[={{a}^{2}}\,\,({{\sin }^{2}}x+{{\cos }^{2}}x)+{{b}^{2}}({{\cos }^{2}}x+{{\sin }^{2}}x)\] |
| \[={{a}^{2}}+{{b}^{2}}\] |
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