Q. 125.0( 2 Votes )

Prove the following:



OR

Prove the following:


Answer :

Given that x (0, 1)

To Prove:


Proof: Take Right Hand Side (RHS),



Substitute x = tan2 θ in the above equation. We get



We know that,


Squaring on both sides,





We know that by trigonometric identity, sin2 θ + cos2 θ = 1.



Also, by trigonometric identity, cos 2θ = cos2 θ – sin2 θ.



And, cos-1(cos 2θ) = 2θ



RHS = θ …(i)


We had assumed x = tan2 θ


If x = tan2 θ


x = (tan θ)2


√x = tan θ


θ = tan-1 √x


So, substituting this value of θ in equation (i), we get


RHS = tan-1 √x


RHS = LHS


Hence, proved.


OR


We are given with a trigonometric equation.


To Prove:


Proof: Take Left Hand Side (LHS),


Let


…(i)


And let


…(ii)


Let us also find sin x and cos y.


We know the trigonometric identity,


sin2 x + cos2 x = 1


We need to find the value of sin x. So,


sin2 x = 1 – cos2 x



Substituting the value of cos x from (i),






…(iii)


Also,


sin2 y + cos2 y = 1


We need to find the value of cos y. So,


cos2 y = 1 – sin2 y



Substituting the value of sin y from (ii),






…(iv)


Now, we have the values of sin x, cos x, sin y and cos y.


Using trigonometric identity,


sin (x + y) = sin x cos y + cos x sin y


Substituting values of sin x, cos x, sin y and cos y from eq. (i), (ii), (iii) and (iv) respectively.






Substituting values of x and y,



LHS = RHS


Hence, proved.


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