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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]