Prove if this is true or false 1-sinϴ • cosϴ •tanϴ=cos^2ϴ and cos^2ϴ • tanϴ= sinϴ

Question

Its two questions Prove if this is true or false 1-sinϴ • cosϴ •tanϴ=cos^2ϴ and cos^2ϴ • tanϴ= sinϴ 

Answer 1

I will use T to represent the angle Theta.

 

The first statement is proven to be true as follows:

 

1 - sin T * cos T * tan T =

 

1 - sin T * cos T * sin T/ cos T =

 

1 - sin T* sin T  =             <--- cosines cancel

 

1 - sin T^2 =

 

cos ^2 T  <---  trig identity sin^2 + cos^2 = 1 , which means 1 - sin^2 = cos^2

 

-------------------------------------------------------------------

 

The second one is false.

 

Let T=30 = pi/6

 

sin 30 = 1/2

cos 30 = sqrt3/2

tan 30 = sqrt(3)/3

 

cos(2T) = cos 60 = 1/2

 

Left side is 1/2*sqrt(3)/3 = sqrt(3)/6

 

Just for the fun of it, we can find where

the identity holds for specific values of T.

 

Using identities, the left side becomes:

 

  cosT^2 * sinT/cosT = cosT*sinT

 

In order for this identity to hold:

  cosT*sinT = sinT

 

 cosT * sin T - sin T = 0

 

 sin T (cos T - 1) = 0

 

sin T = 0  or cos T  - 1 = 0

 

sin T = 0 ---> T is an integral multple of 2*pi.

That is 2*k*pi where K is an integer.

Specifically ..., -3*pi, -2*pi, -pi, 0, pi, 2*pi, 3*pi, ...

is where the sine function is zero.

 

cos T - 1 = 0 ---> cos T = 1

The cosine is 1 on EVEN multiples of pi, 

namely  ..., -4*pi, -2*pi, 0, 2*pi, 4*pi, ...

 

combining these two solutions produces

integral multiples of pi.