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
1) To simplify the given expression (sin2t) (cos5t) + (cos2t) (sin5t), we can use the trigonometric identity for the sine of the sum of two angles:
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
In this case, let a = 2t and b = 5t. Then, we have:
(sin2t) (cos5t) + (cos2t) (sin5t) = sin(2t + 5t) = sin(7t)
So the simplified expression is sin(7t).
2) To solve the equation {1/(sec x - 1)} - {1/(sec x +1)} = 2 cot^2 x, we can use the trigonometric identity for the secant function:
sec^2(x) = 1 + tan^2(x)
First, let's convert the given equation to use secant and tangent functions:
{1/(sec x - 1)} - {1/(sec x +1)} = 2 cot^2 x
Now, we can replace sec^2(x) and tan^2(x) using the trigonometric identity mentioned above:
{1/(1 + tan^2(x) - 1)} - {1/(1 + tan^2(x) + 1)} = 2 cot^2 x
Simplifying further, we get:
{1/tan^2(x)} - {1/(1 + tan^2(x) + 1)} = 2 cot^2 x
Now, let's work on simplifying the denominators:
1/tan^2(x) = cot^2(x)
1/(1 + tan^2(x) + 1) = 1/(2 + tan^2(x)) = 1 - tan^2(x)/[(2 + tan^2(x))]
Substituting these simplifications into the equation, we have:
cot^2(x) - [1 - tan^2(x)/{(2 + tan^2(x))}] = 2 cot^2 x
Expanding the denominator and combining like terms, we get:
cot^2(x) - 1 + tan^2(x)/(2 + tan^2(x)) = 2 cot^2 x
Multiplying through by the denominator (2 + tan^2(x)), we have:
cot^2(x)(2 + tan^2(x)) - (2 + tan^2(x)) + tan^2(x) = 2 cot^2 x(2 + tan^2(x))
Expanding and rearranging terms, we get:
2cot^2(x) + 2cot^2(x)tan^2(x) - 2 - 2tan^2(x) + tan^2(x) = 4cot^2(x) + 2cot^2(x)tan^2(x)
Simplifying, we have:
3cot^2(x) - tan^2(x) - 2 = 0
Now, we can use the trigonometric identity for cotangent:
cot^2(x) = 1 - tan^2(x)
Substituting this back into the equation, we have:
3(1 - tan^2(x)) - tan^2(x) - 2 = 0
Expanding and rearranging terms, we get:
3 - 3tan^2(x) - tan^2(x) - 2 = 0
Combining like terms, we have:
-4tan^2(x) + 1 = 0
Solving for tan^2(x), we get:
tan^2(x) = 1/4
Therefore, the values of x that satisfy the equation are x = arctan(1/2) and x = π + arctan(1/2) (or x = π - arctan(1/2) depending on the interval of x).