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Question
Prove that (sin a + cosec a) 2 + (cos a + sec a) 2 >= 9
Prove that (sin a + cosec a) 2 + (cos a + sec a) 2 >= 9
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Solution
sin a + cosec a) 2 + (cos a + sec a) 2
= Sin²a+Cosec²a+2sinacoseca. + Cos²a+sec²a+2cosaseca
Sin²a+Cosec²a+2sina×1/Sina. + Cos²a+sec²a +2 cosa×1/cosa
Sin²a+Cos²a+ 2×1 +2×1 + sec²a+Cosec²a
1+2+2+(1+tan²a)+(1+cot²a)
7+tan²a + cos²a
7+2
9
Hence proved sin a + cosec a) 2 + (cos a + sec a) 2 >= 9
Tan²a+Cot²a =2
sin a + cosec a) 2 + (cos a + sec a) 2
= Sin²a+Cosec²a+2sinacoseca. + Cos²a+sec²a+2cosaseca
Sin²a+Cosec²a+2sina×1/Sina. + Cos²a+sec²a +2 cosa×1/cosa
Sin²a+Cos²a+ 2×1 +2×1 + sec²a+Cosec²a
1+2+2+(1+tan²a)+(1+cot²a)
7+tan²a + cos²a
7+2
9
Hence proved sin a + cosec a) 2 + (cos a + sec a) 2 >= 9
Tan²a+Cot²a =2
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