Question

Identify transformations of trigonometric expressions prove the following identities $\dfrac{{{{\sin }^2}2a + 4{{\sin }^4}a - 4{{\sin }^2}a{{\cos }^2}a}}{{4 - {{\sin }^2}2a - 4{{\sin }^2}a}} = {\tan ^4}a$

Answer 1

Hint:In the given question the RHS is given in ${\tan ^4}a$ so we have to try to convert the LHS part into ${\tan ^4}a$ or $\dfrac{{{{\sin }^4}a}}{{{{\cos }^4}a}}$. For this apply $\sin 2a = 2\sin a\cos a$ or ${\sin ^2}2a = 4{\sin ^2}a{\cos ^2}a$ try to change $2a$ angles in $a$ . After that apply ${\sin ^2}a + {\cos ^2}a = 1$ to proceed the result .


Complete step-by-step answer:
As in the RHS it is given that ${\tan ^4}a$ so we have to try to convert the LHS part into ${\tan ^4}a$ or $\dfrac{{{{\sin }^4}a}}{{{{\cos }^4}a}}$ .
For this we have to apply trigonometric transformation in the LHS part .
From LHS
$\dfrac{{{{\sin }^2}2a + 4{{\sin }^4}a - 4{{\sin }^2}a{{\cos }^2}a}}{{4 - {{\sin }^2}2a - 4{{\sin }^2}a}}$
So we know that $\sin 2a = 2\sin a\cos a$ , apply this is in the numerator part , hence we get ,$\dfrac{{{{\left( {2\sin a\cos a} \right)}^2} + 4{{\sin }^4}a - 4{{\sin }^2}a{{\cos }^2}a}}{{4 - {{\sin }^2}2a - 4{{\sin }^2}a}}$
$\dfrac{{4{{\sin }^2}a{{\cos }^2}a + 4{{\sin }^4}a - 4{{\sin }^2}a{{\cos }^2}a}}{{4 - {{\sin }^2}2a - 4{{\sin }^2}a}}$
Now $4{\sin ^2}a{\cos ^2}a$ will cancel out ,
$\dfrac{{4{{\sin }^4}a}}{{4 - {{\sin }^2}2a - 4{{\sin }^2}a}}$
$\dfrac{{4{{\sin }^4}a}}{{4(1 - {{\sin }^2}a) - {{\sin }^2}2a}}$
Now in denominator , we know that ${\sin ^2}a + {\cos ^2}a = 1$ or ${\cos ^2}a = 1 - {\sin ^2}a$ apply this in the denominator
$\dfrac{{4{{\sin }^4}a}}{{4{{\cos }^2}a - {{\sin }^2}2a}}$
Now we know that $\sin 2a = 2\sin a\cos a$ and on squaring ${\sin ^2}2a = 4{\sin ^2}a{\cos ^2}a$
$\dfrac{{4{{\sin }^4}a}}{{4{{\cos }^2}a - 4{{\sin }^2}a{{\cos }^2}a}}$

Take $4{\cos ^2}a$ common in the denominator ,
$\dfrac{{4{{\sin }^4}a}}{{4{{\cos }^2}a(1 - {{\sin }^2}a)}}$
$\dfrac{{4{{\sin }^4}a}}{{4{{\cos }^2}a.{{\cos }^2}a}}$

$\dfrac{{4{{\sin }^4}a}}{{4{{\cos }^4}a}} = {\tan ^4}a$
= RHS
Proved

Note:Students should remember trigonometric formulas, identities and transformation formulas for solving these types of problems.For these types of problems first we have to approach the solution by analysing the R.H.S of the given expression, Use the suitable identities , formula and simplify the L.H.S part to prove the given expression.