Use C to denote cos(x), and express 1+tan^2(x)=______ in terms of C. Similarly, express (1-sinx)(1+sinx)=____ in terms of C.

Question

Hint: For the first part use the definition of the tangent function. For the second part use the third binomial formula:

(a+b)(a-b)=a^2-b^2

 

Can you please show clear steps on how you got this? I want to master this as soon as possible.

Answer 1

1+tan²(x)=sec²(x) is a trig identity

sec(x)=1/cos(x) is a trig identity

substitute

1+tan²(x)=(1/cos[x])²

1+tan²(x)=1/cos²(x)

 

sin²(x)+cos²(x)=1 is a trig identity

subtract sin²(x) from both sides to get...

cos²(x)=1-sin²(x)

(1-sinx)(1+sinx)=1-sinx+sinx-sin²(x)

(1-sinx)(1+sinx)=1-sin²(x)

1-sin²(x)=cos²(x) from the above identity