Question

cos4theta- sin4theta=1 - 2sin2theta​

Answer 1

cos4θ−sin4θ=1−2sin2θ

Left Side :

cos4θ−sin4θ

=(cos2θ−sin2θ)(cos2θ+sin2θ)

=(cos2θ−sin2θ).1

=cos2θ−sin2θ

=1−sin2θ−sin2θ

=1−2sin2θ

=

Right Side

Answer 2

$\huge\underline\mathrm{GIVEN:-}$

• cos⁴$\theta$- sin⁴$\theta$=1 - 2sin²$\theta$

$\huge\underline\mathrm{SOLUTION:-}$

$\large\rm\red{\star\: Taking\: L.H.S. \: \star }$

$\longmapsto$$\sf{ (cos^2\theta - sin^2\theta )(cos^2\theta + sin^2\theta )}$【∴ Using Identity, (a² - b²) = (a+ b)(a -b)】

$\longmapsto$$\sf{ (cos^2\theta - sin^2\theta ) \times 1}$

（ ∴ Using Identity, Sin² $\theta$+ cos²$\theta$= 1）

$\longmapsto$$\sf{ 1- sin^2\theta - sin^2\theta }$（ ∴ From Identity 1) , Cos² $\theta$= 1 - sin² $\theta$）

$\longmapsto$$\large\sf\red{ 1 -2\: sin^2\theta }$

$\rm\red{ L.H.S. \iff R.H.S.}$

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