Simplify the expression \$"tan x"("cosx" + "cot x")/("sec x" + "tanx")\$ using reciprocal identities.

The given expression is

\$"tan" "x"("cosx" + "cot x")/("sec x" + "tanx")\$

The formulas are

\$"tan"\theta = ("sin"\theta)/("cos"\theta), "sec"\theta = 1/("cos"\theta), "cot"\theta = ("cos"\theta)/ ("sin"\theta)\$

Substitute these into the expression:

⇒ \$ (sin x / cos x) ( cos x + cos x / sin x ) / ( 1/cos x + sin x / cos x ) \$

After simplify the expression

⇒ \$ (sin x / \cancel(cos x) * \cancel(cos x) + \cancel(sin x) / \cancel(cos x) * \cancel(cos x) / \cancel(sin x) ) / ( (1 + sin x) / cos x) \$

⇒ \$ (sin x + 1 ) / ( (1 + sin x) / cos x ) \$

⇒ \$ ( \cancel(sin x + 1 ) (cos x)) / ( \cancel (1 + sin x)) \$

⇒ \$ cos x \$

So, the simplified expression is \$cos x \$.

This option does not match the simplified expression cosx. It seems to be an expression involving subtraction of 1 from sinx, which is not equivalent to cosx. Therefore, option A is incorrect

-1 + sinx

This option doesn’t match the simplified expression cosx. It suggests an expression involving the tangent of twice the angle x, which is not equivalent to cosx. Therefore, option B is incorrect

tan2x

This option correctly matches the simplified expression. The simplified expression is indeed cosx, as derived using reciprocal identities. Therefore, option C is correct

cosx

This option suggests that the simplified expression is equal to cot2x. However, the expression we simplified does not match the form of cot2x. So, option D is incorrect

cot2x

Option

C

Can you summarize what you’ve understood in the above steps?