Question

Please give me answer with explanation

Answer 1

you have to remember all the trigonometrical functions

Answer 2

$\underline{\underline{\mathfrak{\Large{Solution: }}}}$




$\underline{\textsf{\large{4.}}}$


$\underline {\textsf{To Prove !}} \\ \\ \sf \implies cosec \: \theta \sqrt{1 \: - \: cos^2 \theta} \: = \: 1$


$\textsf{Let,} \\ \\ \sf \implies cosec \: \theta \sqrt{1 \: - \: cos^2 \theta} \: = \: k \\ \\ \sf \implies cosec \: \theta \sqrt{ {sin}^{2} \theta \: + \: \cancel{{cos}^{2} \theta }\: - \: \cancel{ {cos}^{2} \theta} } \: = \: k \\ \\ \sf \implies cosec \: \theta \: \times \: sin \: \theta \: = \: k \\ \\ \sf \implies \frac{1}{ \cancel{sin \: \theta}} \: \times \: \cancel{sin \: \theta} \: = \: k \\ \\ \sf \: \: \therefore \: \: \:k \: \: = \: 1 \\ \\ \underline{\textsf{Proved !! }}$


$\textsf{Trigonometric Identity used : } \\ \\ \sf \implies sin^2 \: \theta \: + \: cos^2 \: \theta \: = \: 1 \\ \\ \textsf{And,} \\ \\ \sf \implies cosec \: \theta \: = \: \dfrac{1}{sin \: \theta}$



$\underline{\textsf{\large{5.}}}$


$\underline{\textsf{To Find : }} \\ \\ \sf = cot^2 \theta \: - \: \dfrac{1}{sin^2 \theta} \\ \\ \sf = cot^2 \theta \: - \: \dfrac{1}{sin^2 \theta} \\ \\ \sf = \dfrac{cos^2 \theta}{sin^2 \theta} \: - \: \dfrac{1}{sin^2 \theta} \quad \left\{ cot^2 \theta \: = \: \dfrac{cos^2 \theta}{sin^2 \theta} \right\} \\ \\ \sf = \dfrac{ {cos}^{2} \theta\: - \: 1 }{ {sin}^{2} \theta} \\ \\ \sf = \dfrac{ \cancel{ {cos}^{2} \theta} \: - \: {sin}^{2} \theta \: - \: \cancel{ {cos}^{2} \theta}}{ {sin}^{2} \theta } \quad \left \{ {sin}^{2} \theta \: + \: {cos}^{2} \theta \: = \: 1 \right \} \\ \\ \sf = \dfrac{ - \cancel{{sin}^{2}} \theta }{ \cancel{{sin}^{2} \theta}} \\ \\ \sf = - 1$



$\underline{\textsf{Another Method : }} \\ \\ \sf = cot^2 \theta \: - \: \dfrac{1}{sin^2\theta } \\ \\ \sf = cot^2 \theta \: - \: cosec^2 \theta \quad \left\{ cosec^2 \theta \: = \: \dfrac{1}{sin^2 \theta} \right\} \\ \\ \sf = -( cosec^2 \theta \: - \: cot^2 \theta) \\ \\ \sf = -1 \quad \left\{cosec^2 \theta \: - \: cot^2 \theta \: = \: 1 \right\}$