Trig Identities! Math problem! Please help!?

Question

prove the following:

(1/ 1+ sinx ) + ( 1/ 1- sinx )= 2Sec^2 x

Also...

what is an equivalent expression for

A. 1- sin^2x

B. 1+ tan^2x

Answer 1

(1/ 1+ sinx ) + ( 1/ 1- sinx )= 2Sec^2 x

(1+1)/[(1+sinx)(1-sinx)] = 2sec^2x

2/(1-sin^2x) = 2sec^2x

2/cos^2x = 2sec^2x

2sec^2x = 2sec^2x

recall sin^2x + cos^2x = 1 and secx = 1/cosx

also A = cos^2x and B = 2tanx/sin2x

Answer 2

(1/1 + sinx) + (1/1 - sinx) = 2sec²x

= [(1 - sinx) + (1 + sinx)]/(1 - sin²x)

= 2/(1 - sin²x)

Using identity sin²x + cos²x = 1:

= 2/(sin²x + cos²x - sin²x)

= 2/cos²x

= 2sec²x

Again, using identity sin²x + cos²x = 1:

a.)

cos²x = 1 - sin²x

b.)

Knowing that tan²x = sin²x/cos²x, divide everything by cos²x:

sin²x/cos²x + cos²x/cos²x = 1/cos²x

tan²x + 1 = sec²x

Answer 3

1                       1

--------------  +    ------------   ?= 2Sec^2(x)

1+sin(x)          1-sin(x)

1                       1                 2

--------------  +    ------------   ?=  -----------

1+sin(x)          1-sin(x)          cos^2(x)

1-sin(x)           1+sin(x)            2

--------------  + ---------------   ?=  -----------

1-sin^2(x)      1-sin^2(x)        cos^2(x)

1-sin(x) + 1+sin(x)            2

----------------------------   ?=  -----------

1-sin^2(x)                      cos^2(x)

2                                     2

----------------------------   ?=  -----------

1-sin^2(x)                      cos^2(x)

2                                     2

----------------------------   ?=  -----------     I'll prove you can do this step next

cos^2(x)                        cos^2(x)

A. Knowing sin^2(x)+cos^2(x)=1 from the pyth. thm. Subtract sin^2(x) from both sides to produce

1-sin^2(x) = cos^2(x) which is justification in the last step of my first derivation.

B. sin(x)/cos(x) = tan(x)

tan^2(x) + 1 =

sin^2(x)

------------ + 1 =

cos^2(x)

sin^2(x) + cos^2(x)

----------     -------------  =

cos^2(x)   cos^2(x)

sin^2(x) + cos^2(x)

---------------------------- =

cos^2(x)

1

-------------- =

cos^2(x)

sec^2(x)