Question

化简下列各题：（1）sin2αsin2β+cos2αcos2β?12cos2αcos2β；（2）tan(x+π4)+tan(x?π4)；（3）sin3

化简下列各题：（1）sin2αsin2β+cos2αcos2β?12cos2αcos2β；（2）tan(x+π4)+tan(x?π4)；（3）sin3α+4cos2α+3sinα?4cos3α?4sin2α+5cosα．

Answer 1

（1）sin2αsin2β+cos2αcos2β?

1

2

cos2αcos2β
=（1-cos²α）（1-cos²β）+cos²αcos²β?

1

2

(2cos2α?1)(2cos2β?1)
=1-cos²α-cos²β+2cos²αcos²β?2cos2αcos2β+cos2β+cos2α?

1

2

=

1

2

（2）tan(x+

π

4

)+tan(x?

π

4

)
=

tanx+1

1?tanx

+

tanx?1

1+tanx

sinx+cosx

cosx?sinx

+

sinx?cosx

cosx+sinx

(sinx+cosx)2+(sinx?cosx)(cosx?sinx)

(cosx?sinx)(cosx+sinx)

=2tan2x
（3）

sin3α+4cos2α+3sinα?4

cos3α?4sin2α+5cosα

=

3sinα?4sin3α+4(1?2sin2α)+3sinα?4

4cos3α?3cosα?4sin2α+5cosα

=tanα