Question

$\sf \: if \: tanx - cotx = \boxed{\red{\dfrac{119}{60}}} \\ \green{\rm \: where \: x \: is \: between \: 0\degree and\:90\degree..}\\ \pink{\bf \: Find: \large\boxed{\blue{\: sin(x) + cos(x)..}}} \: \\ \\ \bf \: full \: solution \: plzz..$
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Answer 1

||✪✪ GIVEN ✪✪||

tanx - cotx = (119/60) where x is between 0° and 90°..

|| ★★ TO FIND ★★ ||

• (sinx + cosx) = ?

|| ✰✰ ANSWER ❶✰✰ ||

Putting cotx = (1/tanx) we get,

⇒ tanx - (1/tanx) = (119/60)

Taking LCM and cross - Multiply now,

⇒ 60(Tan²x - 1) = 119tanx

⇒ 60tan²x - 119tanx - 60 = 0

Splitting The Middle Term now, we get ,

➾ 60tan²x - 144tanx + 25tanx - 60 = 0

➾ 12tanx(5tanx - 12) + 5(5tanx - 12) = 0

➾ (5tanx - 12)(12tanx + 5) = 0

Putting both Equal to Zero now, we get,

➾ tanx = (12/5) or, (-5/12) .

Since 0° < x < 90°.

➾ tanx = (12/5)

___________________________

Now, we know that tanA = (Perpendicular / Base)

So,

⟼ tanx = (12/5) = p/b

we get, p = 12 , b = 5 .

Using Pythagoras Theoram Now, we get:-

➺ h² = p² + b²

➺ h² = (12)² + (5)²

➺ h² = 144 + 25

➺ h² = 169

➺ h = 13

So,

☞ sinx = (Perpendicular /Hypotenuse) = (12/13)

☞ cosx = (Base / Hypotenuse) = (5/13)

___________________________

Putting Both Values Now, we get :-

☛ sinx + cosx

☛ (12/13) + (5/13)

☛ (17/13) (Ans.)

___________________________

|| ✰✰ ANSWER ❷✰✰ ||

➼ (tanx - cotx) = 119/60

Putting tanx = (sinx/cosx) and cotx (cosx/sinx) we get,

➼ (sinx/cosx) - (cosx/sinx) = 119/60

Taking LCM now,

➼(sin²x - cos²x) /(cosxsinx) = 119/60

Taking (-1) Common From Numerator now,

➼ (-1)(cos²x - sin²x) / (cosxsinx) = 119/60

Putting (cos²A - sin²A) = cos2A in Numerator now,

➼ (-cos2x) /(cosxsinx) = 119/60

Multiply and divide by 2 in Numerator and denominator now,

➼ 2(-cos2x) / (2cosxsinx) = 119/60

Putting 2sinA*cosA = sin2A in Denominator now,

➼ (-cos2x)/(sin2x) = 119/120

➼ cot2x = (- 119/120)

➼ Tan2x = (- 120/119)

⟪As, 119, 120 and 169 are Pythagoras Triplets Now.⟫

So,

➼ sin2x = 120/169

Hence,

☛ Ans = √[1 + (120/169)] = (17/13).

______________________________

Answer 2

$\huge{\underline{\textrm{\color{grey}{AnswEr:—}}}}$

$\longrightarrow \:{ \tt{ \frac{17}{13} }}$

$\rule{200}2$

$\huge{\underline{\textrm{\color{grey}{ ExplainaTion:—}}}}$

→ cotx = (1/tanx)

So,

→tanx - (1/tanx) = (119/60)

•Taking LCM and cross - Multiply now,

»60(Tan²x - 1) = 119tanx

» 60tan²x - 119tanx - 60 = 0

•Splitting The Middle Term now, we get ,

→ 60tan²x - 144tanx + 25tanx - 60 = 0

→ 12tanx(5tanx - 12) + 5(5tanx - 12) = 0

→ (5tanx - 12)(12tanx + 5) = 0

•Putting both Equal to Zero ;

→ tanx = (12/5) or, (-5/12) .

• 0° < x < 90°.

→ tanx = (12/5)

→ tanA = (Perpendicular / Base)

tanx = (12/5) = p/b

, p = 12 , b = 5 .

•Using Pythagoras Theorem:—

»h² = p² + b²

» h² = (12)² + (5)²

» h² = 144 + 25

» h² = 169

»h = 13

Now,

sinx = (Perpendicular /Hypotenuse) =

(12/13)

cosx = (Base / Hypotenuse) =

(5/13)

•Putting the above values:-

⟩ sinx + cosx

⟩(12/13) + (5/13)

•°• 17/13

$\rule{200}2$