Question

Cot(45-A)=tan2A+sec2A


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Answer 1

cot(45 -A) = cot(90 - (45 + A)) = tan(45 + A) = tan2A + sec2A

Explanation:

cot(45 -A) = tan2·(A) + sec2·(A)

We have;

tan2(A) + sec2(A) = sin2(A)/cos2(A) + 1/cos2(A) = (sin2(A) + 1)/cos2(A)

= (sin²(A) + cos²(A) + 2·sin·(A)cos(A))/(cos²A - sin²A)

= (sin(A) + cos(A))²/((cos(A) -sin(A))(cos(A) +sin(A)))

= (sin(A) + cos(A))/((cos(A) -sin(A)))

= (cos(A)(1 + sin(A)/cos(A)))/(cos(A)(1 - sin(A)/cos(A)))

= (1 + tan(A))/(1 - tan(A))

= (tan 45 + tan(A))/(1 - tan(45)·tan(A)) = tan(45 + A)

cot(π/2 - x) = tan(x)

= cot(π/2 - (45 + A)) = tan(45 + A)

π/2 = 90°

= cot(90 - (45 + A)) = tan(45 + A)

cot(45 - A) = tan(45 + A) = tan(2A) + sec(2A).