Question

If tan theta = n tan alpha and sin theta is equal to m sin alpha then prove that cos square theta is equal to m square minus one divided by n square minus one​

Answer 1

Given: tan theta = n tan (alpha), sin theta = m sin (alpha)

To find: Prove that cos^2 (theta) = m^2 - 1 / n^2 - 1

Solution:

• Now we have given that sin theta = m sin (alpha)

                 sin theta / sin alpha = m       ...............(i)

• Now we have given tan theta = n tan (alpha).

• We can write it as:

                 sin theta / cos theta = n ( sin alpha / cos alpha )

                 sin theta / sin alpha = n ( cos theta / cos alpha )

• Putting (i) in above, we get:

                 m = n ( cos theta / cos alpha )

                 cos alpha = n/m (cos theta )  ...............(ii)

• From (i), we can write it as:

                  sin alpha = sin theta / m  ....................(iii)

• From (ii) and (iii), we get:

                  sin^2 alpha + cos^2 alpha = sin^2 theta / m^2 + n^2/m^2 (cos^2 theta )

                 sin^2 theta / m^2 + n^2/m^2 (cos^2 theta ) = 1

                 sin^2 theta + n^2(cos^2 theta ) = m^2

                 n^2(cos^2 theta ) + 1 - cos^2 theta = m^2

                 n^2 cos^2 theta - cos^2 theta = m^2 - 1

                 cos^2 theta (n^2 - 1) =  m^2 - 1

                 cos^2 theta =  m^2 - 1 / n^2 - 1  Hence proved.

Answer:

        So in solution part, we proved that cos^2 theta =  m^2 - 1 / n^2 - 1 .