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Question Number 57345 by behi83417@gmail.com last updated on 02/Apr/19

$$1)\mathbf{if}:\mathbf{sinx}+\mathbf{tgx}=1,\mathbf{then}:{\mathbf{sin}}^4\mathbf{x}+{\mathbf{tg}}^4\mathbf{x}=?$$$$2)\mathbf{if}:\mathbf{sinx}+\mathbf{tgx}=2,\mathbf{then}:\mathbf{sin}4\mathbf{x}+\mathbf{tg}4\mathbf{x}=?$$$$3.\mathbf{if}:\mathbf{sinx}+\mathbf{tgx}=3,\mathbf{then}:\frac{\mathbf{sin}4\mathbf{x}}{{\mathbf{sin}}^4\mathbf{x}}+\frac{\mathbf{tg}4\mathbf{x}}{{\mathbf{tg}}^4\mathbf{x}}=?$$

Answered by tanmay.chaudhury50@gmail.com last updated on 03/Apr/19

$$sinx+\frac{sinx}{cosx}=a$$$$\frac{2t}{1+t^2}+\frac{2t}{1+t^2}\times \frac{1+t^2}{1-t^2}=a$$$$\frac{2t}{1+t^2}+\frac{2t}{1-t^2}=a$$$$\frac{2t-2t^3+2t+2t^3}{1-t^4}=a$$$$\frac{4t}{1-t^4}=a$$$$a-{at}^4-4t=0$$$${at}^4+4t-a=0$$$$t=tan\left(\frac{x}{2}\right)$$$$whena=1t=0.249$$$$whena=2t=0.475$$$$whena=3t=0.631$$$$restoftheproblemcanbesolvedby$$$$putting\frac{2t}{1+t^2}forsinxand\frac{2t}{1-t^2}fortanx$$$$thenputvalueoft$$

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