Question

if X cosecteta=1 and y secteta=1 find x²+y²​

Answer 1

Answer:

given,X cosecA=1,and y secA=1

now,X=1/cosecA→X=sinA

And,y =1/secA→y=cosA

now,

x²+y²=sin²A+cos²A

=1

Answer 2

Question :-

$\sf{ \red{if }\: x \: \csc \theta \: = 1 \: \: \green{ and} \: y \: \sec \theta = 1} \\ \sf{ \orange{then \: find \:} {x}^{2} + {y}^{2} }$

Answer :-

→ x ² + y ² = 1

Formula used :-

$</p><p> \red{\boxed{ \green\star}}\: \sin \theta = \frac{1}{ \csc \: \theta} \\ \\ \green{ \boxed{\orange{ \star}}} \: \cos \theta \: = \frac{1}{ \sec \theta} \\ \\ \pink{ \boxed{ \green{ \star } }} \: \: { \sin}^{2} \theta \: + { \cos}^{2} \theta \: = 1$

Solution :-

Given that

$\rightarrow \sf{x \: \csc \theta \: = 1} \\ \\ \rightarrow \sf{ \: x = \frac{1}{ \csc \theta} } \\ \\ \rightarrow \sf{ x \: = \sin \theta} \\ \\ \bf{and }\: \\ \: \\ \rightarrow \sf{y \: \sec \theta \: = 1 }\\ \\ \rightarrow \sf{ y \: = \frac{1}{ \sec \theta} } \\ \\ \rightarrow \sf{y \: = \cos \theta}$

Therefore ,

$\boxed{\implies} \: \: \: \: \: \sf{ {x}^{2} + {y}^{2} } \\ \\ \boxed{ \implies} \: \: \: \: \sf{ { \sin}^{2} \theta \: + { \cos }^{2} \theta \: = 1} \\ \\ \therefore \boxed{ \sf{ {x}^{2} + {y}^{2} = 1}}$

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Some trigonometric formula :-

$\star \: { \sin}^{2} \theta \: + { \cos}^{2} \theta \: = 1 \\ \\ \star \: { \sec}^{2} \theta \: - { \tan }^{2} \theta \: = 1 \\ \\ \star \: { \csc }^{2} \theta - { \cot }^{2} \theta \: = 1 \\ \\ \star \: \sin \theta \: = \frac{1}{ \csc \theta} \\ \\ \star \: \tan \theta = \frac{1}{ \cot \theta}$

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