Question

Упростите выражение(нужен полный ответ) ​

Answer 1

5

$\sin( \alpha ) \cos( \alpha ) ( { \cos }^{2} \alpha - { \sin }^{2} \alpha ) = \sin( \alpha ) \cos( \alpha ) \times \cos( 2\alpha ) = \\ = \frac{1}{2} \times 2 \sin( \alpha ) \cos( \alpha ) \cos( 2\alpha ) = \frac{1}{2} \sin(2 \alpha ) \cos( \2alpha ) = \\ = \frac{1}{2} \times ( \frac{1}{2} \times 2) \sin( 2\alpha ) \cos( 2\alpha ) = \\ = \frac{1}{4} \sin( 4\alpha )$

6

$\frac{ \sin( 4\alpha ) }{ { \cos}^{4} \alpha - { \sin}^{4 } \alpha } = \\ = \frac{ \sin(4 \alpha ) }{( { \cos}^{2} \alpha - { \sin}^{2} \alpha )( { \cos }^{2} \alpha + { \sin}^{2} \alpha ) } = \\ = \frac{ \sin(4 \alpha ) }{ \cos(2 \alpha ) \times 1 } = \frac{ 2\sin( 2\alpha ) \cos(2 \alpha ) }{ \cos( 2\alpha ) } = 2 \sin( 2\alpha )$

7

$\sin( \frac{\pi}{4} - \alpha ) \cos( \frac{\pi}{4} - \alpha ) = \\ = (\sin( \frac{\pi}{4} ) \cos( \alpha ) - \sin( \alpha ) \cos( \frac{\pi}{4} ) ) \times ( \cos( \frac{\pi}{4} ) \cos(\alpha) + \sin( \frac{\pi}{4} ) \sin( \alpha )) = \\ = ( \frac{ \sqrt{2} }{2} \cos( \alpha ) - \frac{ \sqrt{2} }{2} \sin( \alpha ) ) \times ( \frac{ \sqrt{2} }{2} \cos( \alpha ) + \frac{ \sqrt{2} }{2} \sin( \alpha )) = \\ = {( \frac{ \sqrt{2} }{2} \cos( \alpha )) }^{2} - {( \frac{ \sqrt{2} }{2} \sin( \alpha )) }^{2} = \frac{1}{2} { \cos }^{2} \alpha - \frac{1}{2} { \sin }^{2} \alpha = \frac{1}{2} \cos( 2\alpha )$

8

$\frac{2tg(1.5 \alpha )}{1 + {tg}^{2}(1.5 \alpha ) } = \frac{2tg(1.5 \alpha )}{ \frac{1}{ { \cos }^{2} (1.5\alpha) } } = \\ = 2 \times \frac{ \sin( 1.5\alpha ) }{ \cos( 1.5\alpha ) } \times { \cos}^{2}(1.5 \alpha ) = \\ = 2 \sin( 1.5\alpha ) \cos(1.5 \alpha ) = \sin( 3\alpha )$