Prove that cos 2-8-cos 23 -8-cos 25 -8-cos 27 -8-2

To prove cos^2π/8+cos^23π/8+cos^25π/8+cos^27π/8=2 Proof: LHS=cos^2π/8+cos^23π/8+cos^2( π 3π/8+ )+cos^2( π π/8+ ) =cos^2π/8+cos^23π/8+cos^2(3π/8 )+cos^2( π/8 ) ...

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Byju's Answer

Standard XI

Mathematics

Basic Trigonometric Identities

Prove that co...

Question

Prove that $c o s^{2} \frac{π}{8} + c o s^{2} \frac{3 π}{8} + c o s^{2} \frac{5 π}{8} + c o s^{2} \frac{7 π}{8} = 2$

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Solution

To prove
$c o s^{2} \frac{π}{8} + c o s^{2} \frac{3 π}{8} + c o s^{2} \frac{5 π}{8} + c o s^{2} \frac{7 π}{8} = 2$
Proof:
$L H S = c o s^{2} \frac{π}{8} + c o s^{2} \frac{3 π}{8} + c o s^{2} (π - \frac{3 π}{8} +) + c o s^{2} (π - \frac{π}{8} +) = c o s^{2} \frac{π}{8} + c o s^{2} \frac{3 π}{8} + c o s^{2} (\frac{3 π}{8}) + c o s^{2} (\frac{π}{8}) = 2 (c o s^{2} \frac{π}{8} + c o s^{2} \frac{3 π}{8}) = 2 (c o s^{2} \frac{π}{8} + c o s^{2} (\frac{π}{2} - \frac{π}{8})) = 2 (c o s^{2} \frac{π}{8} + s i n^{2} \frac{3 π}{8}) = 2 = R H S$

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Prove that cos2π8+cos23π8+cos25π8+cos27π8=2

To prove cos2π8+cos23π8+cos25π8+cos27π8=2 Proof: LHS=cos2π8+cos23π8+cos2(π−3π8+)+cos2(π−π8+)=cos2π8+cos23π8+cos2(3π8)+cos2(π8)=2(cos2π8+cos23π8)=2(cos2π8+cos2(π2−π8))=2(cos2π8+sin23π8)=2=RHS

Prove that $c o s^{2} \frac{π}{8} + c o s^{2} \frac{3 π}{8} + c o s^{2} \frac{5 π}{8} + c o s^{2} \frac{7 π}{8} = 2$