The value of the expression ( log 8(1+2 cos 2 θ)3 / 4+ log 2( sin θ)1 / 4/ log 16( operatornamecosec 3 θ)) (whenever defined) is equal to

Question

The value of the expression ( log 8(1+2 cos 2 θ)3 / 4+ log 2( sin θ)1 / 4/ log 16( operatornamecosec 3 θ)) (whenever defined) is equal to (A)  log 7 6 ⋅ log 6 7 (B) (√2+1) cot (5 π/8) (C)  log 7 6 ⋅ log 6((1/7)) (D) (√2-1) tan (3 π/8)

Tardigrade Admin · Accepted Answer

E =((1/4) log 2(1+2 cos 2 θ)+(1/4) log 2 sin θ/-(1)4 log 2 sin 3 θ) =( log 2( sin θ+2 cos 2 θ sin θ)/- log 2 sin 3 θ)=( log 2( sin θ+ sin 3 θ- sin θ)/- log 2 sin 3 θ)=-1 Now, (A) log 7 6 ⋅ log 6 7=1 (B) (√2+1) cos (5 π/8)=(√2+1)(-(√2-1))=-1 (D) (√2-1) tan (3 π/8)=(√2-1)(√2+1)=1 Hence, options (B) and (C) are correct

Q. The value of the expression $\frac{l o g _{8} ( 1 + 2 c o s 2 θ ) ^{3/4} + l o g _{2} ( s i n θ ) ^{1/4}}{l o g _{16} ( cosec 3 θ )}$ (whenever defined) is equal to

$lo g_{7} 6 \cdot lo g_{6} 7$

$(2 + 1) cot \frac{5 π}{8}$

$lo g_{7} 6 \cdot lo g_{6} (\frac{1}{7})$

$(2 - 1) tan \frac{3 π}{8}$

Q. The value of the expression log16​(cosec3θ)log8​(1+2cos2θ)3/4+log2​(sinθ)1/4​ (whenever defined) is equal to

log7​6⋅log6​7

(2​+1)cot85π​

log7​6⋅log6​(71​)

(2​−1)tan83π​

Q. The value of the expression $\frac{l o g _{8} ( 1 + 2 c o s 2 θ ) ^{3/4} + l o g _{2} ( s i n θ ) ^{1/4}}{l o g _{16} ( cosec 3 θ )}$ (whenever defined) is equal to

$lo g_{7} 6 \cdot lo g_{6} 7$

$(2 + 1) cot \frac{5 π}{8}$

$lo g_{7} 6 \cdot lo g_{6} (\frac{1}{7})$

$(2 - 1) tan \frac{3 π}{8}$