if tanx+cotx=2 then the value of tan^5x + cot^10x is
If tanx + cotx = 2
sinx/cosx + cosx/sinx = 2
(sin^2 x + cos^2 x)/(sinxcosx) = 2
1/(sinxcosx) = 2
2sinxcosx= 1
sin 2x = 1
2x = 90° for 0 ≤ x ≤ 180°
x = 45° or π/4 radians
we know tan 45° = 1 and cot 45° = 1
then tan^5 x + cot^10 x
= 1^5 + 1^10
= 2
There are other solutions for x
they are x + k(180°) , where k is an integer
but all such angles will fall in either quadrant I or III and both the tangent and cotangent would be +1
so tan^5 x + cot^2 x = 2
To find the value of tan^5x + cot^10x, we can start by expressing tanx and cotx in terms of sine and cosine.
Recall that tanx = sinx/cosx and cotx = cosx/sinx.
Using these expressions, we can rewrite the given equation tanx + cotx = 2 as sinx/cosx + cosx/sinx = 2.
To simplify this equation, let's find a common denominator. Multiplying both terms by cosx*sinx, we get (sin^2x + cos^2x) / (cosx*sinx) = 2.
Since sin^2x + cos^2x = 1, the equation simplifies to 1 / (cosx*sinx) = 2.
Multiplying both sides by (cosx*sinx), we get 1 = 2*cosx*sinx.
Next, let's make use of the trigonometric identity sin(2x) = 2*sinxcosx. Rearranging the equation, we have 1 = sin(2x).
Now, let's solve for x. Taking the inverse sine of both sides, we get 2x = arcsin(1).
Since the range of arcsin(x) is -π/2 to π/2, the value of arcsin(1) is π/2.
Therefore, 2x = π/2.
Simplifying, we have x = π/4.
Now, let's substitute this value into tan^5x + cot^10x.
tan^5(π/4) + cot^10(π/4)
Since tan(π/4) = 1 and cot(π/4) = 1, we have:
1^5 + 1^10
Simplifying, we get:
1 + 1
Thus, the value of tan^5x + cot^10x is 2.
To find the value of tan^5x + cot^10x when tanx + cotx is equal to 2, we need to manipulate the given equation and use appropriate trigonometric identities.
Let's start by squaring the equation tanx + cotx = 2:
(tanx + cotx)^2 = 2^2
tan^2x + 2tanxcotx + cot^2x = 4
Now, we can use the trigonometric identity tan^2x + 1 = sec^2x and cot^2x + 1 = cosec^2x:
(sec^2x - 1) + 2tanxcotx + (cosec^2x - 1) = 4
sec^2x + cosec^2x + (2tanxcotx - 2) = 4
sec^2x + cosec^2x + 2(tanxcotx - 1) = 4
(sec^2x + cosec^2x) + 2(tanxcotx - 1) = 4
(sec^2x + cosec^2x) + 2(tanxcotx - tan^2x - 1) = 4
(sec^2x + cosec^2x) + 2(tanxcotx - tan^2x) = 6
Notice that sec^2x + cosec^2x is equal to (1/cos^2x) + (1/sin^2x), which simplifies to (sin^2x + cos^2x) / (sin^2x * cos^2x). We know from the Pythagorean identity that sin^2x + cos^2x = 1, so the numerator becomes 1.
1/(sin^2x * cos^2x) + 2(tanxcotx - tan^2x) = 6
1/(sin^2x * cos^2x) + 2tanxcotx - 2tan^2x = 6
Now, let's simplify the expression tan^5x + cot^10x. Recall that tan^5x = (tanx)^5 and cot^10x = (cotx)^10.
tan^5x + cot^10x can be rewritten as (tanx)^5 + (cotx)^10.
Since we have an expression involving only tanx and cotx, let's express tan^5x and cot^10x in terms of tanx and cotx.
Using the trigonometric identities:
tan^2x = sec^2x - 1
tan^4x = (tan^2x)^2 = (sec^2x - 1)^2 = sec^4x - 2sec^2x + 1
tan^5x = tan^4x * tanx = (sec^4x - 2sec^2x + 1) * tanx
cot^2x = csc^2x - 1
cot^4x = (cot^2x)^2 = (csc^2x - 1)^2 = csc^4x - 2csc^2x + 1
cot^8x = (cot^4x)^2 = (csc^4x - 2csc^2x + 1)^2 = csc^8x - 4csc^6x + 6csc^4x - 4csc^2x + 1
cot^10x = cot^8x * cot^2x = (csc^8x - 4csc^6x + 6csc^4x - 4csc^2x + 1) * (csc^2x - 1)
Now that we have expressed tan^5x and cot^10x in terms of secx and cscx, we can substitute these values into the equation:
(tanx)^5 + (cotx)^10 = (sec^4x - 2sec^2x + 1) * tanx + (csc^8x - 4csc^6x + 6csc^4x - 4csc^2x + 1) * (csc^2x - 1)
Simplifying this expression may require further algebraic manipulations and trigonometric identities.