展開する (-cot(x)csc(x)(1+sec(x))-sec(x)tan(x)(1+csc(x)))/((1+sec(x))^2)

解

展開する

\frac{- cot ( x ) csc ( x ) ( 1 + sec ( x ) ) - sec ( x ) tan ( x ) ( 1 + csc ( x ) )}{( 1 + sec ( x ) ) ^{2}}

\frac{- cot ( x ) csc ( x ) - csc ^{2} ( x ) - sec ( x ) tan ( x ) - sec ^{2} ( x )}{1 + 2 sec ( x ) + sec ^{2} ( x )}

解答ステップ

\frac{- cot ( x ) csc ( x ) ( 1 + sec ( x ) ) - sec ( x ) tan ( x ) ( 1 + csc ( x ) )}{( 1 + sec ( x ) ) ^{2}}

(1 + sec (x))^{2} = 1 + 2 sec (x) + sec^{2} (x)

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

拡張

- cot (x) csc (x) (1 + sec (x)) - sec (x) tan (x) (1 + csc (x)) : - cot (x) csc (x) - cot (x) csc (x) sec (x) - sec (x) tan (x) - sec (x) tan (x) csc (x)

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

cot (x) csc (x) sec (x) = csc^{2} (x)

sec (x) tan (x) csc (x) = sec^{2} (x)

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