In the figure above, PQRS is a circle. If PQT and SRT
are straight lines, find the value of x.​

Given:

PQRS is a circle, PQT and SRT  are straight lines.

To find:

The value of x.

Solution:

Since PQRS is a circle, PQT and SRT  are straight lines, therefore, PQRS isa cyclic quadrilateral.

We know that, sum of opposite angles of a cyclic quadrilateral is 180 degrees.

Now, SRT  is a straight line.

            (Linear pair)

              ...(i)

According to the Exterior angle theorem, in a triangle the measure of an exterior angle is equal the sum of the opposite interior angles.

Using exterior angle theorem in triangle QRT, we get

Therefore, the value of x is 103 degrees.

I suspect you mean

1/8 sin(4t ) = 1/2 (cos³(t ) sin(t ) - sin³(t ) cos(t ))

On the right side, pull out a factor of cos(t ) sin(t ):

1/2 (cos³(t ) sin(t ) - sin³(t ) cos(t )) = 1/2 cos(t ) sin(t ) (cos²(t ) - sin²(t ))

Recall the double angle identities for sin and cos :

sin(2t ) = 2 sin(t ) cos(t )

cos(2t ) = cos²(t ) - sin²(t )

Then

… = 1/4 (2 cos(t ) sin(t )) (cos²(t ) - sin²(t ))

… = 1/4 sin(2t ) cos(2t )

… = 1/8 (2 sin(2t ) cos(2t ))

… = 1/8 sin(4t )