Question

The given expression wherever defined simplifies to
sin (270+a)cos (720°-a) -- sin(270° - a)sinº (540° + a)
sin 90+ a)sin(-6) - cos(a-180°)
cot(270° - a)
cosec (450° +a)​

Answer 1

Given:

{ [sin (270+a)cos³ (720°-a) - sin(270° - a)sin³ (540° + a)] /[ (sin 90+ a)sin(-a) - cos²(180°-a)] }   + { cot(270° - a)  / cosec² (450° +a)​ }

To find: Simplified answer.

Solution:

• As we have given the trigonometric forms lets convert it in simplest form.

• Lets solve the first bracket, we get:

               { [sin (270+a)cos³ (720°-a) - sin(270° - a)sin³ (540° + a)] /[ (sin 90+ a)sin(-a) - cos²(180°-a)] }  

• Simplifying it, we get:

              [- cos a cos³ a - cos a sin³ a] / -cos a sin a - cos² a

• Taking -cos a common from numerator and denominator, we get:

             -cos a [ cos³ a + sin³ a] / -cos a [cos a + sina]

• Now using formula a³+b³ = (a+b)(a²+b²-ab)

            (cos a + sin a) (cos² a + sin² a -cos a sin a)/(cos a + sin a)

            (1 - cos a sin a) .........(i)

• Now solving the next bracket we get:

            { cot(270° - a)  / cosec² (450° +a)​ }

            tan a / sec² a

            sin a/ cos a x cos² a/1

            sin a cos a .........(ii)

• Combining i and ii we get:

            (1 - cos a sin a) + sin a cos a

            1

Answer:

         So, the given expression wherever defined simplifies to 1.