Question

Find cos^4A - sin ^4A in terms of cos A​

Answer 1

$\cos^4A-\sin^4A\\\\=(\cos^2A)^2-(\sin^2A)^2\\\\=(\cos^2A+\sin^2A)(\cos^2A-\sin^2A)\\\\=\cos^2A-\sin^2A\quad[\because\ \cos^2A+\sin^2A=1]\\\\=\cos^2A-(1-\cos^2A)\\\\=\cos^2A-1+\cos^2A\\\\=2\cos^2A-1\\\\=2\cos A\cdot\cos A-1$

Answer 2

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$\large\red{\boxed{\sf \cos^{4}a - { \sin }^{4} a = ?}}$

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we know That ,

$\large\green{\boxed{\sf ( {x}^{2} - {y}^{2} ) = (x - y)(x + y)}}$

$\large\green{\boxed{\sf \sin^{2}x + { \cos }^{2} y = 1}}$

Using both Now we get,

$\cos^{4}a - { \sin }^{4} a \\ \\ \Rightarrow \: ( { \cos }^{2} a)^{2} - ({ \sin }^{2} a)^{2} \\ \\ \Rightarrow ({ \cos }^{2} a + { \sin }^{2} a)({ \cos }^{2} a - { \sin }^{2} a) \\ \\ \Rightarrow \: 1 \times ({ \cos }^{2} a - { \sin }^{2} a) \: \\ \\ \Rightarrow \: ({ \cos }^{2} a - (1 - { \cos }^{2} a)) \\ \\ \Rightarrow \: ({ 2\cos }^{2} a - 1)$

$\color {red}\large\bold\star\underline\mathcal{Extra\:Brainly\:Knowledge:-}$

$\boxed{\begin{minipage}{7 cm}Fundamental Trignometric Identities\\ \\ \sin^2\theta+\cos^2\theta=1\\ \\ 1+\tan^2\theta=\sec^2\theta \\ \\ 1+\cot^2\theta=\text{cosec}^2 \, \theta \end{minipage}}$