What number should be subtracted from 4(\sin ^{4} 60^{\circ}+\cos ^{4} 30^{\circ}) -(\tan ^{2} 45^{\circ}-\cot ^{2} 30^{\circ})+\cos ^{2} 45^{\circ}- cosec^{2} 45^{\circ}+\sec ^{2} 60^{\circ} to get 2 ?

SSC CHSL 13 Apr 2021 Shift 3 Paper

Section: Quantitative Aptitude

Question : 67 of 100

Marks: +1, -0

What number should be subtracted from

4(\sin ^{4} 60^{∘}+\cos ^{4} 30^{∘})

-(\tan ^{2} 45^{∘}-\cot ^{2} 30^{∘})+\cos ^{2} 45^{∘}-

cosec^{2} 45^{∘}+\sec ^{2} 60^{∘}

to get 2 ?

Solution:

Let the number subtracted be

x

.
Then,

4(\sin ^{4} 60^{∘}+\cos ^{4} 30^{∘})

-(\tan ^{2} 45^{∘}-\cot ^{2} 30^{∘})

+\cos ^{2} 45^{∘}-cosec^{2} 45^{∘}+\sec ^{2} 60^{∘}

-x=2

⇒ x=4[(\;{√{3}}/{2})^{4}+(\;{√{3}}/{2})^{4}]-[(1)^{2}-(√{3})^{2}]

+(\;{1}/{√{2}})^{2}-(√{2})^{2}+(2)^{2}-2

⇒ \;\; x=4 ×[\;{9}/{16}+\;{9}/{16}]-[1-3]

\;\; +\;{1}/{2}-2+4-2

⇒ \;\; x=4 × \;{9}/{8}+2+\;{1}/{2}+2-2

⇒ \;\; x=\;{9}/{2}+\;{1}/{2}+2

=\;{9+1+4}/{2}=\;{14}/{2}=7

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