We can use trigonometric identities to simplify the left-hand side of the equation and show that it is equal to the right-hand side.
First, we can use the identity cot(a) = 1/tan(a) to rewrite the expression as:
x * (1/tan(a)) * tan(90°+a)
Next, we can use the identity tan(90°+a) = -cot(a) to substitute for the tangent term:
x * (1/tan(a)) * (-cot(a))
Simplifying, we get:
-x
Now, let's simplify the right-hand side of the equation:
tan (90°+a). cot (180°-a) + x.sec (90°+a)
Using the identity tan(90°+a) = -cot(a) and cot(180°-a) = -cot(a), we can substitute for the tangent and cotangent terms:
(-cot(a)) * (-cot(a)) + x * (1/cos(a))
Simplifying this expression, we get:
cot^2(a) + x/sec(a)
Now, we can use the identity sec(a) = 1/cos(a) to substitute for secant:
cot^2(a) + x * cos(a)
Finally, we can use the identity cot^2(a) + 1 = csc^2(a) to further simplify the expression:
csc^2(a) - 1 + x * cos(a)
Substituting for the identity csc(a) = 1/sin(a), we get:
(1/sin^2(a)) - 1 + x * cos(a)
Combining like terms, we get:
(1 - sin^2(a))/sin^2(a) + x * cos(a)
Using the identity sin^2(a) + cos^2(a) = 1 to substitute for sin^2(a), we get:
cos^2(a)/sin^2(a) + x*cos(a)
Using the identity cot(a) = cos(a)/sin(a) to substitute for the cotangent term, we get:
cot^2(a) + x*cot(a)
Now, we can use the identity cot(a) = 1/tan(a) to simplify this expression further:
1/tan^2(a) + x/tan(a)
Combining the fractions, we get:
(x+tan^3(a))/tan^2(a)
Now, we can use the identity tan^2(a) + 1 = sec^2(a) to substitute for the tangent term:
(x+sec^3(a))/sec^2(a)
Finally, we can use the identity sec(a) = 1/cos(a) to substitute for secant:
x*cos(a) + cos^3(a)
This expression is equivalent to the right-hand side of the original equation, which we derived from the left-hand side. Therefore, the original equation is true for all values of a and x.