Question

13. If sinA + cosec A = 3, find
the value of sin²A + cosec²A​

Answer 1

$\large\underline{\sf{Solution-}}$

Given :-

• sinA + cosecA = 3

To Find :-

• The value of sin²A + cosec²A

Identities Used :-

$\red{ \boxed{ \sf{ \: {(x + y)}^{2} = {x}^{2} + {y}^{2} + 2xy}}}$

$\red{ \boxed{ \sf{ \: cosecA = \frac{1}{sinA} }}}$

Calculation :-

Given that,

$\rm :\longmapsto\:sinA + cosecA = 3$

On squaring both sides, we get

$\rm :\longmapsto\:(sinA + cosecA)^{2} = {3}^{2}$

$\rm :\longmapsto\: {sin}^{2}A + {cosec}^{2}A + 2 \: sinA \: cosecA = 9$

$\rm :\longmapsto\: {sin}^{2}A + {cosec}^{2}A + 2 \: \cancel {sinA} \: \times \dfrac{1}{ \cancel{sinA}} = 9$

$\rm :\longmapsto\: {sin}^{2}A + {cosec}^{2}A + 2 \: = 9$

$\rm :\longmapsto\: {sin}^{2}A + {cosec}^{2}A = 9 - 2$

$\rm :\longmapsto\: {sin}^{2}A + {cosec}^{2}A = 7$

Hence,

$\bf :\longmapsto\: {sin}^{2}A + {cosec}^{2}A = 7$

Additional Information:-

Relationship between sides and T ratios

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

tan θ = Opposite Side/Adjacent Side

sec θ = Hypotenuse/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Co-function Identities

sin (90°−x) = cos x

cos (90°−x) = sin x

tan (90°−x) = cot x

cot (90°−x) = tan x

sec (90°−x) = cosec x

cosec (90°−x) = sec x

Fundamental Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1