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if sinA+cosecA=3 find the value of sin2A+cosec2A​

Sagot :

MrRoyal

Answer:

[tex]\sin^2 A + cosec^2A = 7[/tex]

Explanation:

Given

[tex]\sin A+cosec\ A=3[/tex]

Required

Find [tex]\sin^2A + cosec^2A[/tex]

[tex]\sin A+cosec\ A=3[/tex]

Square both sides

[tex](\sin A+cosec\ A)^2=3^2[/tex]

[tex](\sin A+cosec\ A)(\sin A+cosec\ A)=9[/tex]

Open brackets

[tex]\sin^2 A + 2\sin A\ cosec\ A + cosec^2A = 9[/tex]

In trigonometry:

[tex]cosec\ A = \frac{1}{\sin A}[/tex]

So, we have:

[tex]\sin^2 A + 2\sin A *\frac{1}{\sin A} + cosec^2A = 9[/tex]

[tex]\sin^2 A + \frac{2\sin A}{\sin A} + cosec^2A = 9[/tex]

[tex]\sin^2 A + 2 + cosec^2A = 9[/tex]

Collect like terms

[tex]\sin^2 A + cosec^2A = 9-2[/tex]

[tex]\sin^2 A + cosec^2A = 7[/tex]

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