Prove the following identities :
`1 - (sin^(2) A)/ (1 + cos A) = cos A`

Answer

`a to q; b to p; c to q; d to s`

Explanation

a. `9 + 16 = 24 sin (A+B) = 37" " ` (on squaring and adding)
`24sin (A+B) = 12`
`sin (A+B) = (1)/(2) rArr sin C= (1)/(2)`
`" "C = 30^(@) or 150^(@)`
`rArr C = 30^(@)`
b. `(sinA + sin B) ^(2) - sin^(2)C = 3 sinA sin B`
or `sin^(2) A -sin ^(2) C +sin^(2)B = sinA sinB`
or `sin(A+C) sin(A-C) + sin^(2)B= sinA sinB`
or ` sin B[sin(A-C) + sin (A+C)] = sin A sinB `
or `2 sin A cos C = sinA ( as sin B ne 0)`
or ` cos C = 1//2`
or `C = 60^(@)`
c. `2 sinx cos x [ 4 cos ^(4)x - 4 sin ^(4) x ] =1 `
or ` ( sin 2x) -[2(cos^(2)x + sin ^(2)x)] [2 cos^(2)c - 2 sin^(2) x] =1`
or ` ( sin 2 x ) 2xx2 cos 2x = 1`
or ` 2 sin 4x =1 `
or ` sin 4x = (1)/(2) or 4x = 30^(@) or x = 7.5^(@)`.
d. Obviously, `AEOD` is a cyiclic quadrilateral, we have
`angle COD = 120^(@) + 45^(@) = 165^(@)`