2tanx/1 - tan2x + 1/ 2 cos2x -1= cosx + sinx/ cosx - sinx

is proved

Solution:

Given that,

Let us first solve the L.H.S

 --- (1)

By trignometric identities,

By using these in (1) we get,

By definition of tan,

Therefore,

 --- (ii)

By trignometric identities,

By using these in (ii)

 ----- (iii)

We know that,

Similarly,

Also,

Similarly,

Therefore apply these in (iii)

Cancel out (cos x + sin x) on numerator and denominator

Thus L.H.S = R.H.S

Thus proved

area of part 1:

a1=base*height/2=(8+2)*(8-2-2-1)/2=10*3/2=15 ft^2

area of part 2:

a2=length*width=(8+2)*1=10 ft2

area of part 3:

a3=length*width=8*2=16 ft2

total shaded area : a1+a2+a3=41 ft2

total area of the reactangle : a=16*8=128 ft2

total area of the nonshaded region : 128-41=87 ft2