Question

$If \: \ \: sin ^{2 } \theta \: = k\\ \ \textless \ br /\ \textgreater \ \red {Then} \: \: \\ \ \textless \ br /\ \textgreater \ \dfrac{\tan {}^{3} \theta}{1 + \tan {}^{2} \theta } + \dfrac{ \cot {}^{3} \theta }{1 + \cot {}^{2} \theta }$
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Answer 1

Step-by-step explanation:

We have,

sinθ=k,

Since,

1+tan 2 θtan 3 θ + 1+cot 2 θcot 3 θ

= sec 2 θtan 3 θ + csc 2 θcot 3 θ

= cos 2 θ 1cos 3 θsin 3 θ + sin 2 θ 1 sin 3 θcos 3

= cosθsin 3 θ + sinθcos 3 θ

=sinθcosθ(sin 2 θ+cos 2θ)=sinθcosθ

= k 2 −1k

Hope it's helpful for you ✌✌

Answer 2

Step-by-step explanation:

We have,

sinθ=k,

Since,

1+tan 2 θtan 3 θ + 1+cot 2 θcot 3 θ

= sec 2 θtan 3 θ + csc 2 θcot 3 θ

= cos 2 θ 1cos 3 θsin 3 θ + sin 2 θ 1 sin 3 θcos 3

= cosθsin 3 θ + sinθcos 3 θ

=sinθcosθ(sin 2 θ+cos 2θ)=sinθcosθ

= k 2 −1k

Hope it's helpful for you ✌✌