Use identities to simplify each expression.
1) (sin2t) (cos5t) + (cos2t) (sin5t)
2) {1/(sec x - 1)} - {1/(sec x +1)} = 2 cot^2 x
sin a cos b + cos a sin b = sin (a+b)
so sin (7t)
1/(1/cos x -1) - 1/(cos x +1)
cos x/(1 - cos x) - cos x/(1+cos x)
[cos x(1+cos x) - cos x(1-cos x)]/(1-cos^2 x)
[ cos x + cos^2x - cos x +cos^2 x]/sin^2x
2 cos^2 x/sin^2 x
2 cot^2 x
To simplify the given expressions, we can make use of trigonometric identities. Let's go through each expression one by one:
1) (sin2t) (cos5t) + (cos2t) (sin5t)
Using the product-to-sum identity, we can rewrite the given expression as:
sin(2t + 5t) + sin(5t - 2t)
Simplifying further:
sin(7t) + sin(3t)
Now, we can use the sum-to-product identity:
2 sin((7t + 3t)/2) cos((7t - 3t)/2)
Simplifying the angles inside the trigonometric functions:
2 sin(5t) cos(2t)
So, the simplified expression is 2 sin(5t) cos(2t).
2) {1/(sec x - 1)} - {1/(sec x +1)} = 2 cot^2 x
To simplify this expression, we need to start by expressing everything in terms of sin(x) and cos(x). First, let's rewrite sec(x) in terms of sin(x) and cos(x):
sec(x) = 1/cos(x)
So, the given expression becomes:
{1/(1/cos(x) - 1)} - {1/(1/cos(x) + 1)} = 2 cot^2 x
Now, simplify the denominators:
{1/(1 - cos(x)/cos(x))} - {1/(1 + cos(x)/cos(x))} = 2 cot^2 x
Simplifying further:
{1/(1 - cos(x))} - {1/(1 + cos(x))} = 2 cot^2 x
To add the fractions, we need a common denominator:
{(1 + cos(x)) - (1 - cos(x))}/{(1 - cos(x))(1 + cos(x))} = 2 cot^2 x
Simplifying the numerator:
2cos(x) / (1 - cos^2(x)) = 2 cot^2 x
Using the Pythagorean identity, cos^2(x) = 1 - sin^2(x), we get:
2cos(x) / (1 - (1 - sin^2(x))) = 2 cot^2 x
Simplifying further:
2cos(x) / sin^2(x) = 2 cot^2 x
Finally, using the identity cot(x) = cos(x) / sin(x):
2 cot(x) = 2 cot^2 x
Therefore, the simplified expression is 2 cot^2 x.