promath The general solution of the differential equation dy/dx=y/x is given by y=cx where c is any constant. To solve dy/dx = y/x, we will use the separation of variables method. Let us learn how to find the solution of dy/dx=y/x. General Solution of dy/dx=y/x Question: Find the general solution of $\dfrac{dy}{dx}=\dfrac{y}{x}$. Answer: $\dfrac{dy}{dx}=\dfrac{y}{x}$ We can … Read more
The integration of the square root of 1+sin2x is given by ∫√(1+sin2x) dx = cosx-sinx+C where C is an integration constant. In this post, we will learn how to integrate square root of 1+sin2x. Integral of Square Root of 1+sin2x Question: Find the integral of 1+sin2x, that is, find ∫√(1+sin2x) dx. Answer: We will use … Read more
The Maclaurin series expansion of cosx or the Taylor series expansion of cosx at x=0 is given as follows: cosx = $\sum_{n=0}^\infty \dfrac{(-1)^n}{(2n)!}x^{2n}$ = $1-\dfrac{x^2}{2!}+\dfrac{x^4}{4!}-\dfrac{x^6}{6!}+\cdots$ Taylor Series Expansion of Sinx at x=0 Note that the Maclaurin series expansion of f(x)=cosx or the Taylor series of a function f(x) at $x=0$ is given by the following … Read more
The cot(x+y) formula or the identity is given by cot(x+y) = $\dfrac{\cot x \cot y -1}{\cot x + \cot y}$. Here we will prove the formula of cot x+y. Proof of cot(x+y) Formula cot(x+y) = $\dfrac{\cot x \cot y -1}{\cot x + \cot y}$. Proof: Let us recall the following two formulas: sin(x+y) = sinx … Read more
The tan(x-y) is the tangent of the difference of the angles x and y. Tan(x-y) formula/identity is given as follows: tan(x-y) = $\dfrac{\tan x -\tan y}{1+\tan x \tan y}$. In this post, we will prove the formula of tan x-y. Proof of tan(x-y) Formula tan(x-y) = $\dfrac{\tan x -\tan y}{1+\tan x \tan y}$ Proof: We … Read more
The function cosx siny is the product of a cosine function and a sine function. In this post, we will learn how to prove the formula/identity of cosx siny. cosx siny Formula The cosx siny formula is given as follows: cosx siny = $\dfrac{\sin(x+y)-\sin(x-y)}{2}$ Let us now prove the above formula of cos x sin … Read more
The formula or the identity of the product sinx cosy is obtained by the transformation of sums or differences of trigonometric angles. sinx cosy formula is given below. sinx cosy = $\dfrac{\sin(x+y)+\sin(x-y)}{2}$ Proof of sinx cosy Formula Let us now prove the sinx siny formula sinx cosy = $\dfrac{\sin(x+y)+\sin(x-y)}{2}$ Proof: It is known that sin(x+y) … Read more
The function sinx siny is the product of two sine functions sinx and siny. The formula or the identity of sinx siny is given as follows: sinx siny = $\dfrac{\cos(x-y)-\cos(x+y)}{2}$ Proof of sinx siny Formula We will now prove the sinx siny formula sinx siny = $\dfrac{\cos(x-y)-\cos(x+y)}{2}$ Proof: We know that cos(x+y) = cosx cosy … Read more
The general solution of tanx=tany for x is given by x=nπ+y where n= 0, ±1, ±2, ±3, …. In this post, we will learn how to find the general solution of tanx=tany as well as tanθ=tanα. Solutions of tanx = tany are given by x=nπ+y, n ∈ ℤ Let us now solve the trigonometric equation … Read more
The simplification of the expression 1-sec2x is equal to -tan2x. In this post, we will establish 1-sec^2x formula. We have: Let us now prove the above formulas. 1-sec2x Formula The formula of 1-sec2x is given below: $\boxed{1-\sec^2 x=-\tan^2 x}$ Proof: To simplify the expression $1-\sec^2 x$, we will follow the below steps: Step 1: We will … Read more