Question

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

Answer 1

Answer:

$\sin^2 A + cosec^2A = 7$

Explanation:

Given

$\sin A+cosec\ A=3$

Required

Find $\sin^2A + cosec^2A$

$\sin A+cosec\ A=3$

Square both sides

$(\sin A+cosec\ A)^2=3^2$

$(\sin A+cosec\ A)(\sin A+cosec\ A)=9$

Open brackets

$\sin^2 A + 2\sin A\ cosec\ A + cosec^2A = 9$

In trigonometry:

$cosec\ A = \frac{1}{\sin A}$

So, we have:

$\sin^2 A + 2\sin A *\frac{1}{\sin A} + cosec^2A = 9$

$\sin^2 A + \frac{2\sin A}{\sin A} + cosec^2A = 9$

$\sin^2 A + 2 + cosec^2A = 9$

Collect like terms

$\sin^2 A + cosec^2A = 9-2$

$\sin^2 A + cosec^2A = 7$