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[tex]\boxed{\sf 1-sin^2\theta=cos^2theta}[/tex]
[tex]\boxed{\sf cos^2\theta=\dfrac{1}{sec^2\theta}}[/tex]
Solution:-
Let's do
[tex]\\ \sf\longmapsto k=sec^2\theta(1+sin\theta)(1-sin\theta)[/tex]
[tex]\\ \sf\longmapsto k=sec^2\theta(1-sin^2\theta)[/tex]
[tex]\\ \sf\longmapsto k= sec^2\theta(cos^2\theta)[/tex]
[tex]\\ \sf\longmapsto k=sec^2\theta\times \dfrac{1}{sec^2\theta}[/tex]
[tex]\\ \sf\longmapsto k=1[/tex]
[tex]\color{lime}\boxed{\colorbox{black}{Answer : - }}[/tex]
[tex] \sec^{2} θ(1 + \sinθ)(1 - \sinθ) = k[/tex]
[tex] \sec^{2} θ \: (1 - { \sin}^{2} θ) = k[/tex]
[tex] { \sec }^{2} θ \cos^{2} θ = k[/tex]
[tex] \sec ^{2} θ. \frac{1}{ {sec}^{2}θ } = k[/tex]
[tex]1 = k[/tex]
Therefore:
[tex] \color{red}k = 1[/tex]