simplify sin(45-x)

Solution

simplify

sin (4 5^{\circ} - x)

- sin (- \frac{π}{4} + x)

Solution steps

sin (4 5^{\circ} - x)

Rewrite using trig identities

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

Rewrite using trig identities

= \frac{1 - tan ( x )}{sec ( x ) 2}

Factor

\frac{1 - tan ( x )}{sec ( x ) 2} : \frac{- 2 ( tan ( x ) - 1 )}{2 sec ( x )}

= \frac{- ( - 1 + tan ( x ) ) 2}{2 sec ( x )}

Express with sin, cos

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

Rewrite using trig identities

= - sin (- \frac{π}{4} + x)

cos (arcsin (\frac{x}{a}))

2 sin (3 x + π)

cos (3 π x + \frac{π}{2})

cos (6 0^{\circ} - x)

(sec^{2} (π x) - 1)^{3}