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Question
The value of the expression `[(sin^2 22^circ + sin^2 68^circ)/(cos^2 22^circ + cos^2 68^circ) + sin^2 63^circ + cos 63^circ sin 27^circ]` is ______.
Options
3
2
1
0
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Solution
The value of the expression `[(sin^2 22^circ + sin^2 68^circ)/(cos^2 22^circ + cos^2 68^circ) + sin^2 63^circ + cos 63^circ sin 27^circ]` is 2.
Explanation:
Given expression,
`(sin^2 22^circ + sin^2 68^circ)/(cos^2 22^circ + cos^2 68^circ) + sin^2 63^circ + cos 63^circ sin 27^circ`
= `(sin^2 22^circ + sin^2(90^circ - 22^circ))/(cos^2(90^circ - 68^circ) + cos^2 68^circ) + sin^2 63^circ + cos 63^circ sin(90^circ - 63^circ)`
= `(sin^2 22^circ + cos^2 22^circ)/(sin^2 68^circ + cos^2 68^circ) + sin^2 63^circ + cos 63^circ * cos 63^circ` ...`[(∵ sin(90^circ - theta) = cos theta),("and" cos(90^circ - theta) = sin theta)]`
= `1/1 + (sin^2 63^circ + cos^2 63^circ)` ...[∵ sin2θ + cos2θ = 1]
= 1 + 1
= 2
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