Question

"Question49
Prove the identity : sin² θ + 1 / (1+ tan² θ ) = 1
Chapter7,Trigonometric identities Exercise -7A , Page number 314"

Answer 1

Solution :

1) To prove : “ sin² θ + 1 / (1+ tan² θ ) = 1 ”

2) Inference ( Understanding the question, Thinking of the method ) : We shall start from Left hand side ( L. H. S) and proceed till we get R. H. S. We will use trigonometric identities too.

3) Formulas we are going to use :

$\rightarrow$ 1/secθ = cosθ
$\rightarrow$ tan²θ +1 = sec²θ
$\rightarrow$ sin²θ + cos²θ = 1


4) Actual proof :

Firstly, We know that, tan²θ +1 = sec²θ

$\\ \implies Left \: hand \: side \: \\ \implies sin^2 \theta + \frac {1}{1 + tan^2 \theta} \\ \implies (sin^2 \theta) + \frac{1}{sec^2 \theta} \\ \implies ( sin^2 \theta) + (cos^2 \theta) \\ \implies 1 \\ \implies Right \: Hand \: side$

Therefore, We proved the identity sin² θ + 1 / (1+ tan² θ ) = 1

Hope it helps!

Answer 2

Answer :-


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To prove :-

sin² θ + { 1 / ( 1 + tan² θ ) } = 1


Salutation :-

L.H.S = sin² θ + { 1 / ( 1 + tan² θ ) }

= sin² θ + { 1 / sec² θ }

[ • As we know , sec² θ - tan² = 1 , So sec² θ = 1 + tan² θ ]

= sin² θ + cos² θ

[ • We know , cos θ = 1 / sec θ , so 1 / sec² θ = cos² θ ]

= 1

[ • We know the value of sin² θ + cos² θ is 1 ]

And R.H.S = 1

So , L.H.S = R.H.S [ • Hence Proved ]


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