cos x (sec x +cos x csc2x)=csc2x
cosx secx=1
cos^2 *csc^2= ctn^2 x
1+ctn^2=csc^2
multipy left by sin^2/sin^2
1/sin^2 (sin^2+cos^2)=
csc^2 = csc^2
To solve the equation cos x (sec x + cos x csc2x) = csc2x, we need to simplify the expression on the left-hand side (LHS) to match the expression on the right-hand side (RHS).
Step 1: Distribute cos x to both terms in the parentheses:
cos x * sec x + cos x * cos x * csc2x = csc2x
Step 2: Simplify the expression by using trigonometric identities:
Recall that sec x is the reciprocal of cos x (sec x = 1/cos x), and csc x is the reciprocal of sin x (csc x = 1/sin x). Also, we know that sin2x + cos2x = 1.
Substitute sec x and csc x in terms of sin x and cos x:
cos x * (1 / cos x) + cos x * cos x * (1 / (sin2x)) = csc2x
Simplify further by canceling out the cos x terms:
1 + cos x * (cos x / sin2x) = csc2x
Now, let's simplify the expression cos x / sin2x. We know that cos x / sin x = cot x, and cot x is the reciprocal of tan x (cot x = 1/tan x).
Substitute cot x:
1 + cos x * cot2x = csc2x
We also know that cot2x +1 = csc2x, which is a Pythagorean identity.
Step 3: Substitute cot2x + 1 for csc2x:
1 + cos x * cot2x = cot2x + 1
Simplify further:
1 - 1 + cos x * cot2x = cot2x
0 + cos x * cot2x = cot2x
This equation is true for all values of x except when cot2x = 0. When cot2x is zero, the equation is undefined.
So, the solution to the equation is all real numbers x except for x values that make cot2x = 0.