If $10\sin^4\theta + 15\cos^4\theta = 6$ then the value of $\displaystyle \frac{27\csc^6\theta + 8\sec^6\theta}{16\sec^8\theta}$ is:

Step 1: Simplify the given equation

We are given the equation $10\sin^4\theta + 15\cos^4\theta = 6$. To simplify this, we can divide the entire equation by $\cos^4\theta$. This will help us express the equation in terms of $\tan\theta$ and $\sec\theta$.

Step 2: Convert to tangent and secant

Dividing by $\cos^4\theta$ gives $10\frac{\sin^4\theta}{\cos^4\theta} + 15\frac{\cos^4\theta}{\cos^4\theta} = \frac{6}{\cos^4\theta}$. This simplifies to $10\tan^4\theta + 15 = 6\sec^4\theta$. We know that $\sec^2\theta = 1 + \tan^2\theta$, so $\sec^4\theta = (1 + \tan^2\theta)^2$. Substituting this, we get $10\tan^4\theta + 15 = 6(1+\tan^2\theta)^2$.

Step 3: Solve for $\tan^2\theta$

Expand the right side of the equation: $6(1 + 2\tan^2\theta + \tan^4\theta) = 6 + 12\tan^2\theta + 6\tan^4\theta$. Rearranging the terms, we get $4\tan^4\theta - 12\tan^2\theta + 9 = 0$. This is a perfect square trinomial, which can be factored as $(2\tan^2\theta - 3)^2 = 0$. Solving for $\tan^2\theta$, we find $2\tan^2\theta = 3$, so $\tan^2\theta = \frac{3}{2}$.

Step 4: Find $\sec^2\theta$ and $\csc^2\theta$

Now that we have $\tan^2\theta = \frac{3}{2}$, we can find $\sec^2\theta$ using the identity $\sec^2\theta = 1 + \tan^2\theta$. So, $\sec^2\theta = 1 + \frac{3}{2} = \frac{5}{2}$. Similarly, we can find $\csc^2\theta$ using $\csc^2\theta = 1 + \cot^2\theta = 1 + \frac{1}{\tan^2\theta}$. This gives $\csc^2\theta = 1 + \frac{1}{3/2} = 1 + \frac{2}{3} = \frac{5}{3}$.

Step 5: Substitute values into the expression

Substitute the values of $\csc^2\theta = \frac{5}{3}$ and $\sec^2\theta = \frac{5}{2}$ into the given expression. The expression becomes $\frac{27(\frac{5}{3})^3 + 8(\frac{5}{2})^3}{16(\frac{5}{2})^4}$. Calculate the powers: $(\frac{5}{3})^3 = \frac{125}{27}$ and $(\frac{5}{2})^3 = \frac{125}{8}$ and $(\frac{5}{2})^4 = \frac{625}{16}$. Substitute these back: $\frac{27(\frac{125}{27}) + 8(\frac{125}{8})}{16(\frac{625}{16})} = \frac{125 + 125}{625} = \frac{250}{625}$. Finally, simplify the fraction $\frac{250}{625}$ by dividing both numerator and denominator by 125, which gives $\frac{2}{5}$.

More from this chapter

• no of solution of the equation x plus 2 tanx = pie/2 in the interval of 0 to 2 pie
• If sin A + sin B + sin C = cos A + cos B + cos C = 0, prove that sin² A + sin² B + sin² C = cos² A + cos² B + cos² C = 3/2.
• Find the value of tan (π)/(8).
• Prove that =(sin 5 x-2 sin 3 x+sin x)/(cos 5 x-cos x)=tan x
• The sum of all values of θ [0, 2π] satisfying 2sin² θ = cos 2θ and 2cos² θ = 3sin θ is:
• Prove that 3 sin (π)/(6) sec (π)/(3)-4 sin (5 π)/(6) cot (π)/(4)=1
• If cos x=-(3)/(5), x lies in the third quadrant, find the values of other five trigonometric functions.
• Convert 40^(°) 20^() into radian measure.

All Trigonometric Functions solutions →