Simplify the trigonometric expression 1 sin x sec xtanx Veri

Simplify the trigonometric expression

1) (sin x sec x)/tanx

Verify the Idenity

1) tan theta + cot theta = sec theta csc theta

2) (2 sin x cos x)/(sin x + cos x)^2 = 1

3) (1-cos^2x) (1+ cot^2x) = 1

Solution

Simplify the trigonometric expression

1) (sin x sec x)/tanx

write sec x as 1/cos x

=(sin x /cos x)/tanx

=(tan x)/tan x

=1

Verify the Idenity

1) tan theta + cot theta

=(sin theta/costheta) + (cos theta/sin theta)

=(sin²theta +cos²theta)/(costheta sintheta)

=1/(costheta sintheta)

=(1/costheta)(1/sintheta)

= sec theta csc theta

2) (2 sin x cos x)/(sin x + cos x)^2

=(2 sin x cos x)/(sin²x + cos²x +2sinx cosx)

=(2 sin x cos x)/(1 +2sinx cosx)

not equal to 1

3) (1-cos^2x) (1+ cot^2x)

(1-cos²x) (1+ (cos²x/sin²x))

(sin²x) (sin²x+ cos²x)/sin²x

=(sin²x+ cos²x)

=1