Let theta = a

Now ,

sinA/cotA+cosecA
=sinA/(cosA/sinA+1/sinA)
=sinA/{(cosA+1)/sinA}
=sin²A/(1+cosA)
=(1-cos²A)/(1+cosA)
=(1+cosA)(1-cosA)/(1+cosA)
=1-cosA
2+sinA/cotA-cosecA
=2+sinA/(cosA/sinA-1/sinA)
=2+sinA/{(cosA-1)/sinA}
=2+sin²A/(cosA-1)
=2+(1-cos²A)/{-(1-cosA)}
=2-(1+cosA)(1-cosA)/(1-cosA)
=2-(1+cosA)
=2-1-cosA
=1-cosA

Solving Sin Q/cot Q cosec Q=2 sinQ/cotQ-COSECQ

Step 1: Simplify the left-hand side of the equation

The left-hand side of the equation can be simplified using trigonometric identities:

Sin Q/cot Q cosec Q = Sin Q/ (cos Q/ sin Q) (1/sin Q)
= Sin^2 Q/ cos Q

Step 2: Rewrite the right-hand side of the equation

The right-hand side of the equation can be rewritten using common denominators:

2 sin Q/cot Q - COSEC Q
= 2 sin Q/ (cos Q/ sin Q) - 1/sin Q
= 2 sin^2 Q/ cos Q - 1/sin Q

Step 3: Substitute the simplified left-hand side into the equation

Now we can substitute the simplified left-hand side into the equation:

Sin^2 Q/ cos Q = 2 sin^2 Q/ cos Q - 1/sin Q

Step 4: Solve for sin Q

To solve for sin Q, we can multiply both sides of the equation by cos Q sin Q:

Sin^3 Q = 2 sin^2 Q cos Q - cos Q
Sin^3 Q - 2 sin^2 Q cos Q + cos Q = 0
Sin Q (sin^2 Q - 2 cos Q sin Q + 1) = 0

Step 5: Find the solutions for sin Q

Next, we can solve for the roots of the equation:

sin Q = 0 or sin^2 Q - 2 cos Q sin Q + 1 = 0

If sin Q = 0, then Q = 0, 180 degrees.

If sin^2 Q - 2 cos Q sin Q + 1 = 0, then we can use the quadratic formula:

sin Q = [2 cos Q ± sqrt((2 cos Q)^2 - 4)]/2
sin Q = cos Q ± sqrt(cos^2 Q - 1)

Therefore, the solutions for sin Q are:

Q = arcsin(0) = 0, 180 degrees
Q = arcsin(cos Q + sqrt(cos^2 Q - 1))
Q = arcsin(cos Q - sqrt(cos^2 Q - 1))

Step 6: Check the solutions

We can check the solutions by plugging them back into the original equation and verifying that both sides are equal.

For example, if Q = 30 degrees, then:

Sin Q/cot Q cosec Q = Sin 30/(cos 30/ sin 30) (1/sin 30) = 2
2 sin Q/cot Q - COSEC Q = 2 sin 30/(cos 30/ sin 30) - 1/sin 30 = 2

Therefore, Q = 30 degrees is a valid solution.

Conclusion:

In conclusion, we can solve the equation Sin Q/cot Q cosec Q = 2 sinQ/cotQ-COSECQ by simplifying both sides of the equation, substituting the simplified left-hand side into the equation, solving for sin Q, finding the solutions for sin Q,