If 4 \sin ^{2}(2 x-10)^{\circ}=3, 0^{\circ} \leq(2 x-10) \leq 90^{\circ}, then find the value of {\sin ^{4}(x-5)^{\circ}+\cos ^{4}(x-5)^{\circ}}/{1-2 \sin ^{2}(3 x-15)^{\circ} \cos ^{2}(3 x-15)^{\circ}}

SSC CGL 17 Aug 2021 Shift 3 Paper

Question : 71 of 100

Marks: +1, -0

4 \sin ^{2}(2 x-10)^{∘}=3

0^{∘} ≤(2 x-10) ≤ 90^{∘}

, then find the value of

\;{\sin ^{4}(x-5)^{∘}+\cos ^{4}(x-5)^{∘}}/{1-2 \sin ^{2}(3 x-15)^{∘} \cos ^{2}(3 x-15)^{∘}}

Solution:

Given,

4 \sin ^{2}(2 x-10)^{∘}=3

⇒ \;\; \sin ^{2}(2 x-10)^{∘}=\;{3}/{4}

⇒ \;\; \sin (2 x-10)^{∘}=√{\;{3}/{4}}

⇒ \;\; \sin (2 x-10)^{∘}=\;{√{3}}/{2}

⇒ \;\; \sin (2 x-10)^{∘}=\sin 60^{∘}

⇒ \;\; 2 x-10^{∘}=60^{∘}

⇒ \;\; 2 x=70^{∘} ⇒ x=35^{∘}

∴ \;\; \;{\sin ^{4}(x-5)^{∘}+\cos ^{4}(x-5)^{∘}}/{1-2 \sin ^{2}(3 x-15)^{∘} \cos ^{2}}

(3 x-15)^{∘}

=\;{\sin ^{4}(35-5)^{∘}+\cos ^{4}(35-5)^{∘}}/{1-2 \sin ^{2}(3 × 35-15^{∘}) \cos ^{2}}

=\;{\sin ^{4} 30^{∘}+\cos ^{4} 30^{∘}}/{1-2 \sin ^{2} 90^{∘} ⋅ \cos ^{2} 90^{∘}}

=\;{({1}/{2})^{4}+({√{3}}/{2})^{4}}/{1-2 × 1 × 0}=\;{({1}/{2})^{4}+({√{3}}/{2})^{4}}/{1}

=\;{1}/{16}+\;{9}/{16}=\;{10}/{16}=\;{5}/{8} .

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