Trigonometric co-function identities are relationships between the basic trigonometric functions (sine and cosine) based on complementary angles. They also show that the graphs of sine and cosine are identical, but shifted by a constant of 2π.
The identities are extremely useful when dealing with sums of trigonometric functions, as they often allow for use of the Pythagorean identities.
In the diagram above, point P′ is symmetric with point P with respect to the line y=x. Let P=(x,y) be the coordinates of P, then P′=(y,x).
Now, let θ denote the angle formed by OP and the positive direction of the x-axis. Then, since OP′ and the +y-direction also make an angle of θ, the angle formed by OP′ and the +x-direction will be 2π−θ. Hence the trigonometric co-functions are established as follows:
If tan(2π−x)+cot(2π−x)=2, what is value of tanx?
From the trigonometric co-function identities, we know that tan(2π−θ)=cotθ, and cot(2π−θ)=tanθ. Hence we have
tan(2π−x)+cot(2π−x)cotx+tanxtanx1+tanx1+tan2xtan2x−2tanx+1(tanx−1)2⇒tanx=2=2=2=2tanx=0=0=1.□
Find the value of cos2(1∘)+cos2(2∘)+cos2(3∘)+⋯+cos2(90∘).
Hint: Use the complementary angle identities above.
From above, we know that cosθ=sin(90∘−θ), so we can write cos(46∘)=sin(44∘),cos(47∘)=sin(43∘), and so on.
The sum then becomes cos2(1∘)+⋯+cos2(44∘)+cos2(45∘)+sin2(44∘)+⋯+sin2(1∘).
We can group the terms cleverly to obtain (cos2(1∘)+sin2(1∘))+(cos2(2∘)+sin2(2∘))+⋯+(cos2(44∘)+sin2(44∘))+cos2(45∘).
By the Pythagorean identities, sin2x+cos2x=1, so our sum is simply 44⋅1+cos2(45∘)=44.5.□