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How do you verify $8{{\sin }^{2}}x{{\cos }^{2}}x=1-\cos (4x)$ ?

seo-qna
Last updated date: 23rd Feb 2024
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IVSAT 2024
Answer
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Hint: First analyse the left hand side of the equation and check whether it is equal to the right hand side of the equation. This can be done by using the suitable trigonometric identities (formulae) and simplify any one side of the equation.

Complete step by step solution:
Sine, cosine and tangent of an angle are trigonometric ratios. These ratios are also called trigonometric functions. When we plot a graph of the trigonometric ratios with respect to all the real values of an angle, we get a graph that has a periodic property. This means that the graph repeats itself after equal intervals of the angle. Other than the trigonometric ratios sine, cosine and tangent we have other trigonometric ratios called cosecant, secant and cotangent.

All the above six trigonometric ratios (functions) are dependent on each other. There are different properties and identities that relate the trigonometric ratios. The equation that has to be verified is $8{{\sin }^{2}}x{{\cos }^{2}}x=1-\cos (4x)$.
Let us simplify the left hand side of the equation. The left hand side is $8{{\sin }^{2}}x{{\cos }^{2}}x$.We can write this expression as,
$8{{\sin }^{2}}x{{\cos }^{2}}x=2(2\sin x\cos x)(2\sin x\cos x)$ …. (i)

Now, we shall use the identity $\sin 2x=2\sin x\cos x$ and substitute this value in (i).
$8{{\sin }^{2}}x{{\cos }^{2}}x=2\sin 2x\sin 2x=2{{\sin }^{2}}2x$ ….. (ii)
Now, we shall use the identity $1-\cos 2A=2{{\sin }^{2}}A$.
Then, this means that $1-\cos 4x=2{{\sin }^{2}}2x$.
Substitute this value in (ii).
$\therefore8{{\sin }^{2}}x{{\cos }^{2}}x=1-\cos 4x$.

Therefore, the left hand side of the given equation is equal to the right hand side.Hence, the given equation is correct.

Note: It is not compulsory to verify a given equation only by simplifying the left hand side of the equation. You can also simply the right hand side and check whether it results as the same as the left hand side. Or you can also simplify both sides of the equation and check whether they gave the same result. In this case, you may simplify the right hand side by using the same identities and make it equal to the left hand side.