Question

Q Find the value of the expression below when theta = 30°
sin²theta + cosec²theta+ cos²theta – cot²theta
(A) O
(B) 73/2
(C) 1
(D) 2​

Answer 1

Answer:

${sin}^{2} \theta + cosec ^{2} \theta+ cos ^{2} \theta - cot ^{2} \theta \\ \sf \: we \: know \: that \: \\ \bullet sin ^{2} \theta + cos ^{2} \theta = 1 \\ and \\ \bullet \: cosec ^{2} \theta - cot ^{2} \theta = 1 \\ \\ \\ \sf \: therefore \\ {sin}^{2} \theta + cos^{2} \theta+ cosec ^{2} \theta - cot ^{2} \theta \\ = 1 + 1 \\ = 2$

So, OPTION(D) :2 is the correct answer

Formula used:-

▶️sin² theta +cos² theta =1

▶️cosec² theta -cot² theta =1

Answer 2

$\begin{gathered}\begin{gathered}\bf \: Given \:  - \begin{cases} &\sf{\theta \: = 30 \degree} \\ &\sf{</p><p>\begin{gathered} {sin}^{2} \theta + cosec ^{2} \theta+ cos ^{2} \theta - cot ^{2} \theta \end{gathered}</p><p>} \end{cases}\end{gathered}\end{gathered}$

$\begin{gathered}\begin{gathered}\bf \: To\: find\:- \: the \: value \: of \: \begin{cases} &\sf{</p><p>\begin{gathered} {sin}^{2} \theta + cosec ^{2} \theta+ cos ^{2} \theta - cot ^{2} \theta \end{gathered}</p><p>}  \end{cases}\end{gathered}\end{gathered}$

$\large\underline\purple{\bold{Solution :- }}$

We know

$\rm :\implies\:sin30 \degree \: = \dfrac{1}{2}$

$\rm :\implies\:cos 30\degree \: = \dfrac{ \sqrt{3} }{2}$

$\rm :\implies\:cosec30\degree = 2$

$\rm :\implies\:cot30\degree = \sqrt{3}$

Now,

Consider,

$\longrightarrow\rm \: </p><p>\begin{gathered} \tt \: {sin}^{2} \theta + cosec ^{2} \theta+ cos ^{2} \theta - cot ^{2} \theta \end{gathered}</p><p>$

$\pink{ \longrightarrow \: \rm \: Put \: \theta \: = \: 30\degree}$

$\rm :\implies\:</p><p>\begin{gathered} \tt {sin}^{2} 30\degree + cosec ^{2} 30\degree+ cos ^{2} 30\degree - cot ^{2} 30\degree \end{gathered}</p><p>$

$\rm :\implies\: {(\dfrac{1}{2}) }^{2} + {(2)}^{2} + {(\dfrac{ \sqrt{3} }{2} )}^{2} - {( \sqrt{3}) }^{2}$

$\rm :\implies\:\dfrac{1}{4} + 2 + \dfrac{3}{4} - 1$

$\rm :\implies\:\dfrac{1 + 16 + 3 -12 }{4}$

$\rm :\implies\:\dfrac{8}{4}$

$\rm :\implies\:2$

Additional Information :-

$</p><p>\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\sf Trigonometry\: Table \\ \begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec A & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot A & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0\end{array}}}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$