Prove-that-sec-4-x-cosec-4-x-sin-2-x-cos-2-x-sec-4-x-

Question Number 15328 by tawa tawa last updated on 09/Jun/17

Prove that .

\sec^{4} (x) - {cosec}^{4} (x) = \frac{\sin^{2} (x) - \cos^{2} (x)}{\sec^{4} (x)}

Answered by RasheedSoomro last updated on 10/Jun/17

LHS :

= (\sec^{2} x + {cosec}^{2} x) (\sec x - cosec x) (secx + cosec x)

= (\frac{1}{\cos^{2} x} + \frac{1}{\sin^{2} x}) (\frac{1}{\cos x} - \frac{1}{\sin x}) (\frac{1}{\cos x} + \frac{1}{\sin x})

= \frac{1}{\cos^{2} x \sin^{2} x} \times \frac{\sin x - \cos x}{\cos x \sin x} \times \frac{\sin x + \cos x}{\cos x \sin x}

= \frac{1}{\cos^{2} x \sin^{2} x} \times \frac{\sin^{2} x - \cos^{2} x}{\cos^{2} x \sin^{2} x}

= \frac{\sin^{2} x - \cos^{2} x}{\cos^{4} x \sin^{4} x}

Commented by tawa tawa last updated on 09/Jun/17

God bless you sir .

Commented by tawa tawa last updated on 09/Jun/17

That means it is not equal

Commented by RasheedSoomro last updated on 09/Jun/17

I think so .

Commented by RasheedSoomro last updated on 10/Jun/17

Yes you are correct .