Solve the trigonometric equations.
1.) SIN²x(CSC²x-1)
2.) COT x SEC X
3.) COS²[(π/2)-x]/COS X
(Note: These mathematical problems are somewhat tricky, but useful for students as of learning how to their fundamental identities. Let's enjoy solving these problems!)
You have no equations to solve, you probably meant "simplify the expressions"
1. sin^2 x (1/sin^2 x - 1)
= 1 - sin^2 x
= cos^2 x
2. (cosx/sinx) (1/cosx)
= 1/sinx
= cscx
3. recall the property of complementary trig ratios, that is...
sin(π/2 - x) = cosx and cos(π/2 - x) = sinx
sec(π/2 - x) = cscx and .....
tan(π/2 - x) = cotx
notice the names:
sine --co-sine or cosine
tangent -- co-tangent or cotangent
secant --- co-secant or cosecant
e.g. sin (90° - 20°) = sin 70° = cos20°
so your question:
COS²[(π/2)-x]/COS X
= sin^2 x / cosx
= (sinx)(sinx/cosx)
= sinx tanx
Sure! Let's solve the trigonometric equations step by step.
1.) SIN²x(CSC²x-1)
To solve this equation, we need to apply trigonometric identities.
First, let's simplify the expression inside the brackets:
CSC²x - 1 = 1/SIN²x - 1
= (1 - SIN²x) / SIN²x
= COS²x / SIN²x
Substituting this value back into the equation:
SIN²x(COS²x / SIN²x)
= COS²x
So, the solution to the equation is COS²x.
2.) COT x SEC x
Again, we will use trigonometric identities to simplify the expression.
COT x SEC x = (COS x / SIN x) * (1 / COS x)
= 1 / SIN x
Therefore, the solution to the equation is 1 / SIN x, or simply CSC x.
3.) COS²[(π/2) - x] / COS x
Let's simplify the expression step by step.
First, simplify the angle inside the COS function:
(π/2) - x = π/2 - x
Now, using the cosine difference identity, COS(a - b) = COS a * COS b + SIN a * SIN b, we have:
COS²[(π/2) - x] = COS²(π/2) * COS²(-x) + SIN²(π/2) * SIN²(-x)
= SIN²x
Now, substitute SIN²x back into the equation:
SIN²x / COS x
This is equivalent to TAN x.
Therefore, the solution to the equation is TAN x.
I hope this explanation helps! If you have any further questions, feel free to ask.