The variant of one question can be applied to new courses, review classes and exercise classes. Equivalent substitution of conditions or conclusions in mathematical propositions, some propositions are equivalent propositions, and they can be deduced from each other. If the conditions (or conditions) of a proposition are replaced by equivalent conditions (or conclusions), a new proposition can be obtained.

Example 1: Equation (a-b)c2+(c-a)c+(b-c)=0 has two equal real roots. Prove that A, B and C constitute arithmetic progression. This proposition can be rewritten as "If (c-a)2-4(a-b)(b-c)=0, it proves that A, B and C become arithmetic progression." In fact, the equation in the original question has two equal real roots, which is equivalent to (c-a)2-4(a-b)(b-c)=0 in the new question.

The original title can also be changed to: "Let A, B and C be three internal angles of a triangle, and (Sina-Sinb) C2+(Sinc-Sina) C+(Sinb-Sinc) = 0 has two equal real roots. Prove that Sina, Xinbo and SINC are arithmetic progression. " Sine theorem shows that in △ABC, (sinA-sinB) C2+(sinC-sina) C+(sinb-sinc) = 0 is equivalent to (a-b)c2+(c-a)c+(b-c)=0, sina, sinb and sinc are arithmetic progression, a.

Example 2: Let tgα and tgβ be two roots of equation c2+ac+a+ 1=0. It is proved that the condition of (α+β)= 1 remains unchanged, and the conclusion can be changed to prove that sin(α+β)=cos(α+β).

Or change it to "Verify α+β=nπ+, (n is an integer). Using the interrelation in mathematics, there is a dialectical relationship of unity of opposites in mathematics. For example, addition, subtraction, multiplication and division, power sum root, exponent and logarithm, inverse trigonometric function and trigonometric function, sum and difference product and product and difference, etc. This inspires us to make variants according to the reciprocal relationship in mathematics.