Need help proving trig identities?

Question

I'm having trouble trying to prove these trig identities. Help is greatly appreciated! Thanks in advance!

 

1) cot^2θ +1/sec^2θ = cot^2θ

 

2) 1+cosx/sinx = sinx/1 - cosx

 

3) tanθ + cotθ = secθcscθ

Answer 1

1) (cot²+1)/sec²=cot²

    csc²/sec²=cot²

   (1/sin²)/(1/cos²)=cot²

   (1/sin²)(cos²/1)=cot²

   cos²/sin²=cot²

   (1/tan²)=cot²

   (cot²)=(cot²)

 

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

    (1+cos)(1-cos)/(sin)(1-cos)=sin/(1-cos)

    (1-cos²)/(sin)(1-cos)=sin/(1-cos)

    (sin²)/(sin)(1-cos)=(sin)/(1-cos)

    (sin)/(1-cos)=(sin)/(1-cos)

    or you could have simply cross-multiplied !!

 

3) tan+cot=(sec)(csc)

    (sin/cos)+(cos/sin)=(sec)(csc)

    (sin²+cos²)/(sin)(cos)=(sec)(csc)

    1/(sin)(cos)=(sec)(csc)

    (1/sin)(1/cos)=(sec)(csc)

    (csc)(sec)=(sec)(csc)