Question

If tan square theta is equal to 1-k square, show that dec theta + tan cube theta. cosec theta is equal to +-(2-k square) 3/2

Answer 1

Correct Question :

If tan² θ = 1 - k², Show that sec θ + tan³ θ.cosec θ = ( 2 - k² ) ^ ( 3/2 )

Answer :

Given :

tan² θ = 1 - k² ----- Eq( 1 )

From, sec θ + tan³ θ.cosec θ

= sec θ + tan² θ.tan θ.cosec θ

Expressing tanθ.cosecθ in terms of sinθ and cosθ

= sec θ + tan² θ × ( sin θ / cos θ ) × ( 1 / sin θ )

= sec θ + tan² θ × ( 1 / cos θ )

Using inverse of cos θ i.e sec θ we get,

= sec θ + tan² θ × sec θ

Taking sec θ common

= sec θ ( 1 + tan² θ )

Using identity sec² θ = 1 + tan² θ

= sec θ × sec² θ

= sec³ θ

It can be written as

= ( sec² θ ) ^ ( 3/2 )

Using identity sec² θ = 1 + tan² θ

= ( 1 + tan² θ ) ^ ( 3/2 )

From Eq( 1 )

= ( 1 + 1 + k² ) ^( 3/2 )

= ( 2 + k² )^( 3/2 )

Hence shown.