if cotA78EVALUATE 1sinA1sinA1cosA1cosASolutionTo evaluate 1s

if cotA=7/8,EVALUATE (1+sinA)(1-sinA)/(1+cosA)(1-cosA)

Solution

To evaluate :(1+sinA)(1-sinA)/[(1+cosA)(1-cosA)]

Carefully note that I have introduced bracket in the above expression.

The Numerator :

(1+sinA)(1-sinA) =1-(sinA)^2 , as (x+y)(x-y)=(x^2-y^2)

=1-(sinA)^2 =(cosA)^2,         (1),as (sin X)^2+(cosX)^2 =1.

The dinominator:

(1+cosA)(1-cosA)= 1-(cosA)^2=

= (sinA)^2,                           (2), Reasons as given above.

From (1) and (2) . the given expression is equal to (cosA)^2/(sinA)^2 = (cotA)^2=(7/8)^2=cotA=49/64=0.765625 , as the value of cotA is given to be 7/8.

NB:

If the bracket is not used, the expression like ab/cd means abd/c. The multiplication or division , having equal priority (by BODMAS or PEDMAS),the operation on ab/cd can start from left one. ab/cd= (ab)/cd=((ab)/c)d.

So,(1+sinA)(1-sinA)/(1+cosA)(1-cosA)=(1+sinA)(1-sinA)(1-cosA)/(1+cosA) = (cos^2A)(1-cosA)/(1+cosA)= (7/113)(1-7/sqrt113)/(1+7/sqrt113)

=(49/113)(sqrt113 - 7)/(sqrt113 + 7)

=(49/113)(sqrt113-7)^2/(113-49)

=(49/113){113+49-14sqrt113)/64

=(49/113)(162-14sqrt113)/64 =0.089286498.

Finding the value of cosA from cotA=7/8:

cotA = 7/8=> tanA =8/7.cosA = sqrt(1/sec)^2 = sqrt[(1/(1+tanA)^2]= sqrt(1/(1+(8/7)^2) = 7/sqrt113.

Therefore, cosA =7/sqrt113. We used this value above.

Hope this helps, with some extra details.