Use trigonometric identities to transform the left side of the equation into the right side (0 < 𝜃 < 𝜋/2). 1 + cos 𝜃 1 − cos 𝜃 = 1 - (______) = sin2 𝜃

Answer 1

1.1 radians looks like the answer

= about 63 degrees = about 0.35pi radians

skip, scroll down about half way to the "UNLESS" alternative soluiton = 1.1

but 1st is probably false interpretation of the problem

1 -cosx = sin2x

= 1-sin2x= cosx

1- 2sinxcosx = cosx

- 2sinxcosx =cosx-1

-2(+/-sqr(1-cos^2(x)) = cosx -1

square both sides

4(1-cos^2(x)) = cos^2(x) -2cosx +1

4 -4cos^2(x) = cos^2(x) -2cosx+ 1

5cos^2(x) -2cosx =-3

let y = cosx

5y^2 -2y +3 = 0

factor

(5y+3)(y-1) = 0

y = cosx = 1 or -3/5

x = cos^-1(-3/5) or cos^-1(1)

x=0 would be one answer but it's not in the doman 0<x<90 as it doesn't include the end points

cosine of any angle is >0 for 0<x<90

so there is no solution given the domain restrictions

UNLESS more likely you meant

(1+cosx)(1-cosx) = 1-cos^2(x) = sin2x

then

1-(1-sin^2x) = sin^2(x) = sin2x

sin^2(x) = 2sinxcosx

sin^2(x) -2sinxcosx = 0

sinx(sinx -2cosx) = 0

x=0 would be on solution but the domain excludes it

that leaves

sinx = 2cosx

sqr(1-cos^2(x))/2 = 2cosx

square both sides

(1-cos^2(x)/4 = 4cos^2(x)

1-cos^2(x) = 16cos^2(x)

17cos^2(x) = 1

cos^2(x) = 1/17

cosx = sqr(1/17) = 1/sqr17 = sqr17/17 = about 0.2425

(ignore the negative square root as that puts x outside the domain)

x = cos^-1(sqr(1/17))

x = about 75.97 degrees

check the answer

1-cos^2(75.97) = about 1-(.2425)^2 =1-.0588= .9412

sin2(75.97) = sin151.94 = sin28.06 = about .4704

there's an error above somewhere, since .941 doesn't equal .47, not close even for approximations.

find the error(s) and you win 2 points towards a first class plane trip. No refunds.

to a heavily bombed Ukraine combat zone plus a free parachute for a soft landing

but the answer to the problem's question, what goes in the parentheses is cos^2(x).

that was too easy, the problem must have wanted more?

another approach is graph 1-cos^2(x) = sin2x

and look for the x intercepts where 0<x<pi/2

desmos' graphing calculators shows x=0 and x=1.1

1.1 radians = 1.1(180)/pi = about 63.025 degrees

check the answer

1-cos^2(63.025) = about .795

sin2(63.025) = sin 126.05 = about .809

both are about .80 which is close enough given rounding errors

1.1 radians = theta = x = about 63 degrees