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What is the  value of \(? = \frac{{ta{n^2}{{60}^0} - 2si{n^2}{{45}^0}}}{{cos{{24}^0}cos{{37}^0}coses{{53}^0}cos{{60}^0}cosec{{66}^0} + si{n^2}{{60}^0}}}\)
1. 2
2. \(1\frac{4}{{5}}\)
3. 1
4. \(1\frac{3}{{5}}\)

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Correct Answer - Option 4 : \(1\frac{3}{{5}}\)

Given:

\(? = \frac{{ta{n^2}{{60}^0} - 2si{n^2}{{45}^0}}}{{cos{{24}^0}cos{{37}^0}coses{{53}^0}cos{{60}^0}cosec{{66}^0} + si{n^2}{{60}^0}}}\)

Formula:

cosec(90 - a) = sec a

Calculation:

First solve denominator,

cosec 66º = cosec (90º - 24º) = sec 24º

cosec 53º = cosec (90º - 37º) = sec 37º

Then,

= cos24º.cos37º.cosec53º.cos60º.cosec66º + sin260º

= cos24º.cos37º.sec37º.sec24º.cos60º + sin260º

= cos60º + sin260º

= 1/2 + 3/4

= 5/4

Numerator,

= tan260º - 2sin245º

= 3 - 1

= 2

Then,

⇒ ? = 2/(5/4)

⇒ ? = 8/5

∴ [(tan260º - 2sin245º)/(cos24º.cos37º.cosec53º.cos60º.cosec66º + sin260º)] = \(1\frac{3}{{5}}\)

 

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