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Question

Verify that the following are identities:

(1+sinz)(1-sinz)= 1/sec^2 z <---that last part is one over secent squared z

Answer 1

(1 + sinz)(1 - sinz) ------------>(x + y)(x - y) = x^2 - y^2
1 - sin^2(z) -------------> sin^2(z) + cos^2(z) = 1
cos^2(z)
= 1/sec^2(z)

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Answer 2

LHS = (1 + sin z)(1 - sin z)
Use (a + b)(a - b) = a^2 - b^2
= 1 - sin^2 z
= cos^2 z since sin^2 z + cos^2 z = 1
= 1/sec^2 z
= RHS

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Answer 3

Let's first change the right side:
1/ sec² z = cos² z.
Now multiply out the left side
We get
1 - sin² z.
Thus your equation reduces to
1 - sin² z = cos² z,
which we know is an identity.

Answer 4

(1 + sin z)*(1 - sin z) = 1 - (sin z)^2 = (cos z)^2 = 1/(sec z)^2.

Answer 5

LHS
(1 + sin z) (1 - sin z)
1 - sin ² z
1 - (1 - cos ² z)
cos ² z

RHS
1 / (1 / cos ² z)
cos ² z

LHS = RHS as required