express (secA+cotA)(secA-cotA) in terms of "sin" and simplify....

Question

HINT

1)Convert all the terms into sin or cos.

2)Use identity sin^2x+cos^2x=1

Answer 1

The expression expands to sec^2(A)-cot^2(A).

This can be written:

1/cos^2(A)-cos^2(A)/sin^2(A)=1/(1-sin^2(A))-(1-sin^2(A))/sin^2(A).

For convenience let S=sinA, then we have 1/(1-S^2)-(1-S^2)/S^2=1/(1-S^2)-1/S^2+1.

This may be written: (2S^2-1)/(S^2(1-S^2))=(2sin^2(A)-1)/(sin^2(A)(1-sin^2(A))) which is an expression in sinA only.

cos2A=1-2sin^2(A), so 2sin^2(A)-1=-cos2A; sin2A=2sinAcosA, so sin^2(2A)=4sin^2(A)(1-sin^2(A)).

Therefore, we have -4cos(2A)/sin^2(2A)=-4cot(2A)cosec(2A) as another way to write the original expression.