prove that (sin (x)+ sin (3x)+ sina (5x) )/ (cos x +cos 3x + cos 5x) = tan 3x
视频信息
答案文本
视频字幕
Today we will prove a trigonometric identity. We need to show that the fraction with sine terms in the numerator and cosine terms in the denominator equals tangent of 3x. The graph shows both sides are identical, confirming our identity.
The first step in our proof is to rearrange the terms strategically. We group the first and third terms in both numerator and denominator. This creates pairs that we can simplify using sum-to-product formulas. The grouping transforms our expression into a more manageable form.
Next, we apply the sum-to-product formulas to our grouped terms. For sine x plus sine 5x, we get 2 sine 3x cosine 2x. For cosine x plus cosine 5x, we get 2 cosine 3x cosine 2x. Notice that both results contain the common factor 2 cosine 2x.
Now we substitute our sum-to-product results back into the original fraction. The numerator becomes 2 sine 3x cosine 2x plus sine 3x. The denominator becomes 2 cosine 3x cosine 2x plus cosine 3x. We can factor out sine 3x from the numerator and cosine 3x from the denominator, revealing the common factor 2 cosine 2x plus 1.
Now we can cancel the common factor 2 cosine 2x plus 1 from both numerator and denominator. This leaves us with sine 3x over cosine 3x, which by the tangent identity equals tan 3x. Therefore, we have successfully proven that the original expression equals tan 3x. The identity is complete.