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Category: Algebra

Simplify-2-sin-cos-sin-cos-

Question Number 202041 by hardmath last updated on 19/Dec/23 $$\mathrm{Simplify}:\:\:\:\frac{\sqrt{\mathrm{2}}\:−\:\mathrm{sin}\alpha\:−\:\mathrm{cos}\alpha}{\mathrm{sin}\alpha\:−\:\mathrm{cos}\alpha} \\ $$ Answered by cortano12 last updated on 19/Dec/23 $$\:=\:\frac{\sqrt{\mathrm{2}}−\sqrt{\mathrm{2}}\:\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\mathrm{sin}\:\alpha+\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\mathrm{cos}\:\alpha\right)}{\:\sqrt{\mathrm{2}}\:\left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\mathrm{sin}\:\alpha−\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\:\mathrm{cos}\:\alpha\right)} \\ $$$$\:=\:\frac{\mathrm{1}−\mathrm{cos}\:\left(\alpha−\mathrm{45}°\right)}{\mathrm{sin}\:\:\left(\alpha−\mathrm{45}°\right)} \\ $$$$\:=\:\mathrm{csc}\:\left(\alpha−\mathrm{45}°\right)−\mathrm{cot}\:\left(\alpha−\mathrm{45}°\right) \\…

Solve-1-x-1-x-3-1-2-1-x-2-1-x-3-1-2-1-

Question Number 201991 by MATHEMATICSAM last updated on 18/Dec/23 $$\boldsymbol{\mathrm{Solve}} \\ $$$$\left(\frac{\mathrm{1}}{{x}}\:−\:\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:+\:\left(\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\:−\:\frac{\mathrm{1}}{{x}^{\mathrm{3}} }\right)^{\frac{\mathrm{1}}{\mathrm{2}}} \:=\:\mathrm{1} \\ $$ Answered by mr W last updated…

If-and-are-the-roots-of-the-ax-2-2bx-c-0-and-and-are-the-roots-of-Ax-2-2Bx-C-0-for-some-constant-then-prove-that-b-2-ac-a-2-B-2-AC-A-2-

Question Number 202019 by MATHEMATICSAM last updated on 18/Dec/23 $$\mathrm{If}\:\alpha\:\mathrm{and}\:\beta\:\mathrm{are}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\: \\ $$$${ax}^{\mathrm{2}} \:+\:\mathrm{2}{bx}\:+\:{c}\:=\:\mathrm{0}\:\mathrm{and}\:\alpha\:+\:\delta\:\mathrm{and}\:\beta\:+\:\delta\:\mathrm{are} \\ $$$$\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:{Ax}^{\mathrm{2}} \:+\:\mathrm{2}{Bx}\:+\:{C}\:=\:\mathrm{0}\:\mathrm{for}\:\mathrm{some}\: \\ $$$$\mathrm{constant}\:\delta\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that} \\ $$$$\frac{{b}^{\mathrm{2}} \:−\:{ac}}{{a}^{\mathrm{2}} }\:=\:\frac{{B}^{\mathrm{2}} \:−\:{AC}}{{A}^{\mathrm{2}} }\:. \\…