`a)`

`M = \sin^2 42^o + \sin^2 43^o + \sin^2 44^o + \sin^2 45^o + \sin^2 46^o + \sin^2 47^o + \sin^2 48^o`

`= (\sin^2 42^o + \sin^2 48^o) + (\sin^2 43^o + \sin^2 47^o) + (\sin^2 44^o + \sin^2 46^o) + \sin^2 45^o`

`= (\sin^2 42^o + \cos^2 42^o) + (\sin^2 43^o + \cos^2 43^o) + (\sin^2 44^o + \cos^2 44^o) + \sin^2 45^o`

`= 1 + 1 + 1 + ((\sqrt{2})/(2))^2`

`= 3 + (1)/(2)`

`= (7)/(2)`

`b)`

`N = \cos^2 15^o - \cos^2 25^o + \cos^2 35^o - \cos^2 45^o + \cos^2 55^o - \cos^2 65^o + \cos^2 75^o`

`= (\cos^2 15^o + \cos^2 75^o) - (\cos^2 25^o + \cos^2 65^o) + (\cos^2 35^o + \cos^2 55^o) - \cos^2 45^o`

`= (\cos^2 15^o + \sin^2 15^o) - (\cos^2 25^o + \sin^2 25^o) + (\cos^2 35^o + \sin^2 35^o) - \cos^2 45^o`

`= 1 - 1 + 1 - ((\sqrt{2})/(2))^2`

`= 1 - (1)/(2)`

`= (1)/(2)`

`c)`

`A = \cos^2 1^o + \cos^2 2^o + \cos^2 3^o + ... + \cos^2 87^o + \cos^2 88^o + \cos^2 89^o - (1)/(2)`

`= (\cos^2 1^o + \cos^2 89^o) + (\cos^2 2^o + \cos^2 88^o) + ... + (\cos^2 44^o + \cos^2 46^o) + \cos^2 45^o - (1)/(2)`

`= (\cos^2 1^o + \sin^2 1^o) + (\cos^2 2^o + \sin^2 2^o) + ... + (\cos^2 44^o + \sin^2 44^o) + \cos^2 45^o - (1)/(2)`

$= \underbrace{1 + 1 + \dots + 1}_{44 \text{ số 1}} + \frac{1}{2} - \frac{1}{2}$

$= 44$

`-------`

CT:

`1.`

`\sin(90^\circ - \alpha) = \cos \alpha`

`\cos(90^\circ - \alpha) = \sin \alpha`

`2.`

`\sin^2 \alpha + \cos^2 \alpha = 1`

`3.`

`\sin 45^o = \cos 45^o = (\sqrt{2})/(2)`

`\sin^2 45^o = \cos^2 45^o = ((\sqrt{2})/(2))^2 = (1)/(2)`

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