if sec tita is 5/4 verify tan tita/1 plus tan^2 tita= sin tita/cosec tita
If you can't spell theta you might as well just use x.
secθ = 5/4
so,
sinθ = 3/5
cscθ = 5/3
tanθ = 3/4
tanθ/(1+tan^2θ) = sinθ/cscθ
(3/4)/(1+9/16) = (3/5)/(5/3)
(3/4)/(25/16) = 9/25
3/4 * 16/25 = 9/25
12/25 = 9/25
Hmmm. I suspect a flaw in the equation.
So, is it ever true?
tanθ/(1+tan^2θ) = sinθ/cscθ
tanθ/sec^2θ = sin^2θ
sinθ cosθ = sin^2θ
sinθ(cosθ-sinθ) = 0
θ = π/4
if cot x is 3/4 prove that root of sec x-cosec x/ sec x plus cosec x= 1/root 7
if cot x = 3/4
sec x = 5/3
csc x = 5/4
So, plug in the values and see whether it's true.
To verify the given equation, we will start with the left-hand side (LHS) of the equation and simplify it step by step until we reach the right-hand side (RHS). Let's begin:
Given equation: tan(tita)/(1 + tan^2(tita)) = sin(tita)/cosec(tita)
Step 1: Simplify the LHS
We know that tan(tita) = sin(tita)/cos(tita). Replacing tan(tita) in the equation, we get:
sin(tita)/cos(tita) / (1 + (sin(tita)/cos(tita))^2)
Now, let's eliminate the complex fraction by multiplying the numerator and denominator by the conjugate of the denominator:
sin(tita)/cos(tita) * (cos(tita))^2 / (cos(tita))^2 * (1 + (sin(tita)/cos(tita))^2)
Simplifying further:
sin(tita) * cos(tita) / (cos(tita))^2 + sin^2(tita)
Step 2: Simplify the RHS
We know that sin(tita) = 1/cosec(tita). Replacing sin(tita) in the equation, we get:
1/cosec(tita) / cosec(tita)
Multiplying the numerator and denominator by cosec(tita) to get rid of the complex fraction:
1 / cosec(tita) * cosec(tita) / cosec(tita)
Simplifying further:
1/cosec(tita)
Step 3: Simplify the LHS further
Using the trigonometric identity: 1 + cot^2(tita) = cosec^2(tita)
We can rewrite the LHS as:
sin(tita) * cos(tita) / (cos(tita))^2 + sin^2(tita)
= sin(tita) * cos(tita) / (cos(tita))^2 + (sin(tita))^2 * ((cos(tita))^2 / (cos(tita))^2)
= sin(tita) * cos(tita) + (sin(tita))^2 * (cos(tita))^2 / (cos(tita))^2
= sin(tita) * cos(tita) + (sin(tita))^2 * cos^2(tita) / (cos(tita))^2
= sin(tita) * cos(tita) + (sin(tita))^2 * cos(tita)
Step 4: Simplify the LHS even more
Let's combine the terms on the LHS:
(sin(tita) * cos(tita) + (sin(tita))^2 * cos(tita)) / (cos(tita))^2
= sin(tita) * (cos(tita) + (sin(tita)) * cos(tita)) / (cos(tita))^2
= sin(tita) * cos(tita) * (1 + sin(tita)) / (cos(tita))^2
Step 5: Verify the equation
Comparing the simplified LHS and the RHS, we observe that:
LHS = sin(tita) * cos(tita) * (1 + sin(tita)) / (cos(tita))^2
RHS = 1/cosec(tita)
Now, we know that cosec(tita) = 1/sin(tita). Replacing cosec(tita) in the RHS, we get:
RHS = 1/(1/sin(tita))
= sin(tita)
Therefore, LHS = RHS, and the previous equation is verified.