Question

PROVE THAT
cos(θ) / 1 - sin(θ) + cos(θ) / 1 + sin(θ) = 2 sec(θ)

Answer 1

Step-by-step explanation:

$\boldsymbol{ \underline{Solution:-}}$

$\boldsymbol {\underline{ \pink{To \: \: prove \: \: that \: :-}}}$

$\bf \: \dfrac{cos \: \theta}{1 - \: sin \: \theta} + \dfrac{cos \: \theta}{1 + sin \: \theta} = 2 \: sec \: \theta$

Proof :

LHS :

$\sf \implies \purple{ \dfrac{cos \: \theta}{1 - sin \: \theta} + \dfrac{cos \: \theta}{1 + sin \: \theta} }$

$\sf \implies \purple{ \dfrac{cos \: \theta + sin \: \theta \: cos \: \theta + cos \: \theta - sin \: \theta \: cos \: \theta}{ {1}^{2} - {sin}^{2} \theta} }$

$\sf \implies \purple{ \dfrac{2 \: cos \: \theta}{ {cos}^{2} \: \theta} }$

$\sf \implies \purple{2 \left (\dfrac{1}{cos \: \theta} \right)}$

we know that,

1/cos A = sec A

$\sf \implies \purple{2 \: sec \: \theta}$

LHS = RHS

Hence Proved !!

Answer 2

Step-by-step explanation:

Step-by-step explanation:

\boldsymbol{ \underline{Solution:-}}

Solution:−

\boldsymbol {\underline{ \pink{To \: \: prove \: \: that \: :-}}}

Toprovethat:−

\bf \: \dfrac{cos \: \theta}{1 - \: sin \: \theta} + \dfrac{cos \: \theta}{1 + sin \: \theta} = 2 \: sec \: \theta

1−sinθ

cosθ

+

1+sinθ

cosθ

=2secθ

Proof :

LHS :

\sf \implies \purple{ \dfrac{cos \: \theta}{1 - sin \: \theta} + \dfrac{cos \: \theta}{1 + sin \: \theta} }⟹

1−sinθ

cosθ

+

1+sinθ

cosθ

\sf \implies \purple{ \dfrac{cos \: \theta + sin \: \theta \: cos \: \theta + cos \: \theta - sin \: \theta \: cos \: \theta}{ {1}^{2} - {sin}^{2} \theta} } ⟹

1

2

−sin

2

θ

cosθ+sinθcosθ+cosθ−sinθcosθ

\sf \implies \purple{ \dfrac{2 \: cos \: \theta}{ {cos}^{2} \: \theta} }⟹

cos

2

θ

2cosθ

\sf \implies \purple{2 \left (\dfrac{1}{cos \: \theta} \right)} ⟹2(

cosθ

1

)

we know that,

1/cos A = sec A

\sf \implies \purple{2 \: sec \: \theta}⟹2secθ

LHS = RHS

Hence Proved !!