The sum of all values of [ ] [ ] ( 0, 2 ) ( satisfying sin2 2] + cos4 2 =3 4

Question

The sum of all values of $$\theta $$ $$ \in $$ $\left( {0,{\pi  \over 2}} \right) ( satisfying
sin2 2\]\theta $+ cos4 2$\theta $=${3 \over 4}$

Accepted Answer

sin22$$\theta $$ + cos42$$\theta $$ = $${3 \over 4}, $$ $\theta $$ \in $$\left( {0,{\pi  \over 2}} \right) (

\] \Rightarrow $1$-$cos22$\theta $+ cos42$\theta $=${3 \over 4}$$ \Rightarrow $4cos2$\theta $$-$4cos22$\theta $+ 1 = 0$ \Rightarrow $(2cos22$\theta $$-$1)2 = 0$ \Rightarrow $cos22$\theta $=${1 \over 2}$= cos2${{\pi  \over 4}}$$ \Rightarrow $2$\theta $= n$\pi $$ \pm $${\pi  \over 4}$, n$ \in $I$ \Rightarrow $$\theta $=${{n\pi } \over 2} \pm {\pi  \over 8}$$ \Rightarrow $$\theta $=${\pi  \over 8},{\pi  \over 2} - {\pi  \over 8}$Sum of solutions${\pi  \over 2}$

Solutions · 1