Question

Prove that sin square 40 +sin square 50 /cos square 20 +cos square 70 +tan 10 tan 20 tan 60 tan 70 tan 80 =1+root 3

Answer 1

$\mathsf{Given :\;sin^240 + \dfrac{sin^250}{cos^220 + cos^270} + tan10.tan20.tan60.tan70.tan80}$

★  We know that : cosθ = sin(90 - θ)

$:\implies$  cos20 = sin(90 - 20) = sin70

★  We know that : tanθ = cot(90 - θ)

$:\implies$  tan10 = cot(90 - 10) = cot80

$:\implies$  tan20 = cot(90 - 20) = cot70

Substituting the above values in the given question, We get :

$\mathsf{\implies sin^240 + \dfrac{sin^250}{sin^270 + cos^270} + cot80.cot70.tan60.tan70.tan80}$

★  We know that : sin²θ + cos²θ = 1

$:\implies$  sin²70 + cos²70 = 1

$\mathsf{\implies sin^240 + sin^250 + cot80.tan80.cot70.tan70.tan60}$

★  We know that : tanθ.cotθ = 1

$:\implies$  cot80.tan80 = 1

$:\implies$  cot70.tan70 = 1

$\mathsf{\implies sin^240 + sin^250 + tan60}$

★  We know that : sinθ = cos(90 - θ)

$:\implies$  sin40 = cos(90 - 40) = cos50

$\mathsf{\implies cos^250 + sin^250 + tan60}$

$\mathsf{\implies 1 + tan60}\\\\\mathtt{(since\;\;sin^250 + cos^250 = 1)}$

$\bigstar\;\;\mathsf{We\;know\;that : tan60 = \sqrt{3}}$

$\mathsf{\implies 1 + \sqrt{3}}$

Answer 2

Answer:

Given:sin

2

40+

cos

2

20+cos

2

70

sin

2

50

+tan10.tan20.tan60.tan70.tan80

★ We know that : cosθ = sin(90 - θ)

:\implies:⟹ cos20 = sin(90 - 20) = sin70

★ We know that : tanθ = cot(90 - θ)

:\implies:⟹ tan10 = cot(90 - 10) = cot80

:\implies:⟹ tan20 = cot(90 - 20) = cot70

Substituting the above values in the given question, We get :

\mathsf{\implies sin^240 + \dfrac{sin^250}{sin^270 + cos^270} + cot80.cot70.tan60.tan70.tan80}⟹sin

2

40+

sin

2

70+cos

2

70

sin

2

50

+cot80.cot70.tan60.tan70.tan80

★ We know that : sin²θ + cos²θ = 1

:\implies:⟹ sin²70 + cos²70 = 1

\mathsf{\implies sin^240 + sin^250 + cot80.tan80.cot70.tan70.tan60}⟹sin

2

40+sin

2

50+cot80.tan80.cot70.tan70.tan60

★ We know that : tanθ.cotθ = 1

:\implies:⟹ cot80.tan80 = 1

:\implies:⟹ cot70.tan70 = 1

\mathsf{\implies sin^240 + sin^250 + tan60}⟹sin

2

40+sin

2

50+tan60

★ We know that : sinθ = cos(90 - θ)

:\implies:⟹ sin40 = cos(90 - 40) = cos50

\mathsf{\implies cos^250 + sin^250 + tan60}⟹cos

2

50+sin

2

50+tan60

\begin{gathered}\mathsf{\implies 1 + tan60}\\\\\mathtt{(since\;\;sin^250 + cos^250 = 1)}\end{gathered}

⟹1+tan60

(sincesin

2

50+cos

2

50=1)

\bigstar\;\;\mathsf{We\;know\;that : tan60 = \sqrt{3}}★Weknowthat:tan60=

3

\mathsf{\implies 1 + \sqrt{3}}⟹1+

3