Question

Find the value of : sec²10° – cot²80° + sin 15°cos 75° +cos 15° sin 75°/cos θ sin(90°–θ)+sin θ cos(90°–θ).

Answer 1

SOLUTION IS IN THE ATTACHMENT

Trigonometry is the study of the relationship between the sides and angles of a triangle.

Two angles are said to be complementary of their sum is equal to 90° .
θ & (90° - θ) are complementary angles.

An equation involving trigonometry ratios of an angle is called is called a trigonometric  identity, if it is true for all values of the angles involved. For any acute angle θ, we have 3 identities.
i) sin² θ + cos² θ = 1 ,ii) 1 + tan² θ = sec² θ , iii) cot² θ +1 = cosec² θ.

HOPE THIS WILL HELP YOU....

Answer 2

Hi there!

We have,
sec²(10) − cot²(80) + sin(15) cos(75) + cos(15) sin(75)/ cosθ sin(90-θ) + sinθ cos(90-θ) 

Let's divide these into three parts :

♦ sec²(10) − cot²(80)
♦ sin(15) cos(75) + cos(15) sin(75)
♦ cosθ sin(90-θ) + sinθ cos(90-θ) 

Simplifying Each :

♦ sec²(10) − cot²(80) 

= 1/cos²(10) − cos²(80)/sin²(80) 

= 1/sin²(80) − cos²(80)/sin²(80) 

= (1−cos²(80))/sin²(80) 

= sin²(80)/sin²(80) 

= 1 

♦ sin(15) cos(75) + cos(15) sin(75) 

= sin(15+75) 

= sin(90) 

= 1 

♦ cosθ sin(90-θ) + sinθ cos(90-θ) 

= sin [ θ+(90−θ) ]

= sin(90) 

= 1 

Putting all them back together :

= 1 + 1 / 1

= 1 + 1

= 2

[ Thank you! for asking the question. ]
Hope it helps!

[ NOTE : It's was a bit long question... O_O, that's why I've done it in parts ]