Question

-2sin32*cos32/cos26=?

Answer 1

$\frac{-2sin32*cos32}{cos26}= \frac{-sin64}{cos(90-64)}= \frac{-sin64}{sin64}=-1$

Answer 2

We can simplify the expression as follows:

-2sin32cos32/cos26 = -2(sin32/cos26)(cos32)

Using trigonometric identities, we know that sin(2θ) = 2sinθcosθ. We can apply this identity to simplify the first term:

sin32/cos26 = sin(216)/cos(213) = 2sin16cos16/(cos^2(13) - sin^2(13)) = 2sin16cos16/cos26

Substituting this back into our expression, we get:

-2(sin32/cos26)(cos32) = -2(2sin16cos16/cos26)*cos32

Now, we can use the double angle formula for cosine, which is cos(2θ) = cos^2(θ) - sin^2(θ), to simplify cos32:

cos32 = cos(2*16) = cos^2(16) - sin^2(16)

Substituting this back into our expression and simplifying, we get:

-2*(2sin16cos16/cos26)cos32 = -4sin16cos16(cos^2(16) - sin^2(16))/cos26

Finally, we can use the Pythagorean identity for sine and cosine, which is sin^2(θ) + cos^2(θ) = 1, to simplify our expression:

-4sin16cos16*(cos^2(16) - sin^2(16))/cos26 = -4sin16cos16*(1 - 2sin^2(16))/cos26 = -4sin16cos16/cos26 + 8sin^2(16)*sin32/cos26

Therefore, the simplified expression is -4sin16cos16/cos26 + 8*sin^2(16)*sin32/cos26. Note that we could also have used the product-to-sum formula for sine to further simplify the expression, but this form is already quite simple.