Question

If A, B,C are interior angles of triangle ABC show that
Cosec^2 (B+C/2) - Tan^2(A/2) = 1
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Answer 1

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Answer 2

Given :

A + B + C = 180 [ angle sum property ]

cosec² ( B + C ) / 2 - tan² A / 2

A + B + C = 180

= > B + C = 180 - A

cosec² ( B + C ) / 2 - tan² A / 2

= cosec² ( 180 - A ) / 2 - tan² A / 2

= cosec² ( 90 - A/2 ) - tan² A / 2

We know that cosec ( 90 - A ) = sec A

cosec² ( 90 - A/2 ) = sec² A/2

We know that sec²x - tan²x = 1

sec² A/2 - tan² A/2 = 1 [ Proved ]