Question 76

Find$$ (1 - cos^{2}\theta )(cot^{2}\theta + 1) - 1$$.

Solution

${sin^2(} θ) + {cos^}^{2(} θ) = 1$

$1 + {cot^}^{2(} θ) = {cosec^}^{2(} θ)$

${(sin^}^{2(} θ) + {cos^}^{2(} θ) = 1)$

${(sin^}^{2(} θ) = 1 - {cos^}^{2(} θ))$

Now let's substitute:

$((1 - {cos^}^{2(} θ)) (1 + {cot^}^{2(} θ))-1$

$(({sin^}^{2(} θ)) ({cosec^}^{2(} θ)))-1$

The relationship between sin and cosec is:

$cosec = \frac{1}{sin}$

thus

${cosec^}^{2(} θ) = \frac{1}{{sin^}^{2(} θ)}$

and so ( $({sin^}^{2(} θ)) (\frac{1}{{sin^}^{2(} θ)}))-1 = 1-1 = 0$

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