Упростите 1) 1/cos^2(a) – tg^2(a)-sin^2(a) 2) cos^2(a)+ctg^2(a)-1/sin^2(a) 3) 1/cos^2(a)...

Question

Упростите 1) 1/cos^2(a) – tg^2(a)-sin^2(a) 2) cos^2(a)+ctg^2(a)-1/sin^2(a) 3) 1/cos^2(a) – tg^2(a)(cos^2(a)+1) 4) (1+sin^2(a))ctg^2(a) – 1/sin^2(a) 5)sin(a)/(1+cos(a)) + sin(a)/(1-cos(a)) 6)cos(a)/(1+sin(a))+ cos(a)/(1-sin(a))

Answer 1

1)

1/cos^2(a) – tg^2(a)-sin^2(a)=

=1/cos^2(a) – sin^2(a)/cos^2(a))-sin^2(a)=

=[1 – sin^2(a) ]/cos^2(a))-sin^2(a)=

=cos^2(a)/cos^2(a))-sin^2(a)=

=1--sin^2(a)=cos^2(a)

2)

cos^2(a)+ctg^2(a)-1/sin^2(a)=

=cos^2(a)+[cos^2(a)-1]/sin^2(a)=

=cos^2(a)-sin^2(a)]/sin^2(a)=

=cos^2(a)-1 = -sin^2(a)

3)

1/cos^2(a) – tg^2(a)(cos^2(a)+1)=

=1/cos^2(a) – sin^2(a)-sin^2(a)/(cos^2(a)=

=(1 – sin^2(a))/cos^2(a)-sin^2(a)=

=1-sin^2(a)=cos^2(a)

 

Answer 2

4) (1+sin^2(a))ctg^2(a) – 1/sin^2(a)=cos^2/sin^2 +cos^2 – 1/sin^2(a)=

=(cos^2 - 1)/sin^2 +cos^2= -sin^2/sin^2 +cos^2= -1+cos^2=  -sin^2(a)

 

5)sin(a)/(1+cos(a)) + sin(a)/(1-cos(a))=sin(a) *  ((1-cosa)+(1+cosa)) / (1-cos^2)=2/sin(a)

 

6)cos(a)/(1+sin(a))+ cos(a)/(1-sin(a))=cos(a)* ((1-sina)(1+sina)) / (1-sin^2)= 2/cos(a)