Bonsoir,
[tex]\tan\theta=\dfrac{\sin\theta}{\cos\theta}\\\\cosec\ \theta=\dfrac{1}{\sin\theta}\\\\\sin^2\theta+\cos^2\theta=1\Longrightarrow\sin^2\theta=1-\cos^2\theta[/tex]
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[tex]\tan\theta\cos\theta+cosec\ \theta(\sin\theta-\cos^2\theta\sin\theta)\\\\=\dfrac{\sin\theta}{\cos\theta}\cos\theta+\dfrac{1}{\sin\theta}(\sin\theta-\cos^2\theta\sin\theta)\\\\=\dfrac{\sin\theta}{\cos\theta}\cos\theta+\dfrac{1}{\sin\theta}\sin\theta(1-\cos^2\theta)\\\\=\dfrac{\sin\theta\cos\theta}{\cos\theta}+\dfrac{\sin\theta}{\sin\theta}(1-\cos^2\theta)\\\\=\sin\theta+1\times(1-\cos^2\theta)\\\\=\sin\theta+(1-\cos^2\theta)\\\\=\boxed{\sin\theta+\sin^2\theta}[/tex]
