If tan theta =20/21 show that 1-sin theta +cos theta / 1+sin theta+cos theta =3/7
Answers
Answer:
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Step-by-step explanation:
tanθ=
21
20
As 1+tan
2
θ=sec
2
θ
1+
21×21
20×20
=sec
2
θ
1+
441
400
=sec
2
θ
441
841
=sec
2
θ
secθ=
21
29
cosθ=
29
21
(1)
As 1=cos
2
θ+sin
2
θ
1=
29×29
21×21
+sin
2
θ
1=
841
441
+sin
2
θ
1−
841
441
=sin
2
θ
841
400
=sin
2
θ
29
20
=sinθ (2)
1+sinθ+cosθ
1−sinθ+cosθ
(3)
Substituting (1) and (2) in (3)
1+
29
20
+
29
21
1−
29
20
+
29
21
=
1+
29
41
1+
29
1
=
29
70
29
30
=
70
30
=
7
3
LHS=RHS
Step-by-step explanation:
Given,
tan ∅ = 20/21
tan∅ = opposite side/adjacent side = 20/21
let,
BC = 20, AB = 21, AC = ?,
Now,
According to PHYTHAGHAROUS THEROM,
(Hypotenuse)² = (side)² + (side)²
(AC)² = (AB)² + (BC)²
(AC)² = (21)² + (20)²
(AC)² = 441 + 400
(AC)² = 841
AC = √841
AC = √29 × 29
:. AC = 29 units.
Now consider,
Sin∅ = opposite side/hypotenuse = BC/AC = 20/29
Cos∅ =adjacent side/hypotenuse = AB/AC = 21/29
Now consider,
1-sin∅ + cos∅/1+sin∅ + cos∅
= 1-(20/29) + (21/29)/1+(20/29) + (21/29)
= (29-20+21/29) / (29+20+21/29)
= 30/29 / 70/29
= 30/70
= 6/14
= 3/7
:. 1-sin∅ + cos∅/1+sin∅ + cos∅ = 3/7.
__________ Hence Proved ____________
