Question

$prove \: that \: \frac{2}{ \ \sqrt{2 \sqrt{2 + 2cos4x} } } = secx$
​

Answer 1

ANSWER:

To Prove:

$\dfrac{2}{\sqrt{2+\sqrt{2+2\cos4x\:}}}=\sec x$

Solution:

We need to prove that,

$\implies\dfrac{2}{\sqrt{2+\sqrt{2+2\cos4x\:}}}=\sec x$

Taking LHS,

$\implies\dfrac{2}{\sqrt{2+\sqrt{2+2\cos4x\:}}}$

$\implies\dfrac{2}{\sqrt{2+\sqrt{2(1+\cos4x)\:}}}$

We know that,

$\hookrightarrow\cos2\theta=2\cos^2\theta-1$

Similarly,

$\hookrightarrow\cos4\theta=2\cos^22\theta-1$

So,

$\implies\dfrac{2}{\sqrt{2+\sqrt{2(1+\cos4x)\:}}}$

$\implies\dfrac{2}{\sqrt{2+\sqrt{2(1+(2\cos^22x-1))\:}}}$

$\implies\dfrac{2}{\sqrt{2+\sqrt{2(1\!\!\!/\:+2\cos^22x-1\!\!\!/\:)\:}}}$

$\implies\dfrac{2}{\sqrt{2+\sqrt{2(2\cos^22x)\:}}}$

$\implies\dfrac{2}{\sqrt{2+\sqrt{4\cos^22x\:}}}$

So,

$\implies\dfrac{2}{\sqrt{2+2\cos2x\:}}$

$\implies\dfrac{2}{\sqrt{2(1+\cos2x)\:}}$

$\implies\dfrac{2}{\sqrt{2(1+(2\cos^2x-1))\:}}$

$\implies\dfrac{2}{\sqrt{2(1\!\!\!/\:+2\cos^2x-1\!\!\!/\:)\:}}$

So,

$\implies\dfrac{2}{\sqrt{4\cos^2x\:}}$

$\implies\dfrac{2\!\!\!/\:}{2\!\!\!/\:\cos x}$

$\implies\dfrac{1}{\cos x}$

So,

$\implies\bf sec x = RHS$

As, LHS = RHS

HENCE PROVED!!